diff --git a/base/bitarray.jl b/base/bitarray.jl
index 6175a492cac75..1db84cad37a1c 100644
--- a/base/bitarray.jl
+++ b/base/bitarray.jl
@@ -1707,6 +1707,8 @@ map!(::typeof(identity), dest::BitArray, A::BitArray) = copyto!(dest, A)
 for (T, f) in ((:(Union{typeof(&), typeof(*), typeof(min)}), :(&)),
                (:(Union{typeof(|), typeof(max)}),            :(|)),
                (:(Union{typeof(xor), typeof(!=)}),           :xor),
+               (:(typeof(nand)),                             :nand),
+               (:(typeof(nor)),                              :nor),
                (:(Union{typeof(>=), typeof(^)}),             :((p, q) -> p | ~q)),
                (:(typeof(<=)),                               :((p, q) -> ~p | q)),
                (:(typeof(==)),                               :((p, q) -> ~xor(p, q))),
diff --git a/base/bool.jl b/base/bool.jl
index 92a27543d2fbc..297aa1863a64e 100644
--- a/base/bool.jl
+++ b/base/bool.jl
@@ -70,6 +70,74 @@ julia> [true; true; false] .⊻ [true; false; false]
 """
 xor(x::Bool, y::Bool) = (x != y)
 
+"""
+    nand(x, y)
+    ⊼(x, y)
+
+Bitwise nand (not and) of `x` and `y`. Implements
+[three-valued logic](https://en.wikipedia.org/wiki/Three-valued_logic),
+returning [`missing`](@ref) if one of the arguments is `missing`.
+
+The infix operation `a ⊼ b` is a synonym for `nand(a,b)`, and
+`⊼` can be typed by tab-completing `\\nand` or `\\barwedge` in the Julia REPL.
+
+# Examples
+```jldoctest
+julia> nand(true, false)
+true
+
+julia> nand(true, true)
+false
+
+julia> nand(true, missing)
+missing
+
+julia> false ⊼ false
+true
+
+julia> [true; true; false] .⊼ [true; false; false]
+3-element BitVector:
+ 0
+ 1
+ 1
+```
+"""
+nand(x...) = ~(&)(x...)
+
+"""
+    nor(x, y)
+    ⊽(x, y)
+
+Bitwise nor (not or) of `x` and `y`. Implements
+[three-valued logic](https://en.wikipedia.org/wiki/Three-valued_logic),
+returning [`missing`](@ref) if one of the arguments is `missing`.
+
+The infix operation `a ⊽ b` is a synonym for `nor(a,b)`, and
+`⊽` can be typed by tab-completing `\\nor` or `\\veebar` in the Julia REPL.
+
+# Examples
+```jldoctest
+julia> nor(true, false)
+false
+
+julia> nor(true, true)
+false
+
+julia> nor(true, missing)
+false
+
+julia> false ⊽ false
+true
+
+julia> [true; true; false] .⊽ [true; false; false]
+3-element BitVector:
+ 0
+ 0
+ 1
+```
+"""
+nor(x...) = ~(|)(x...)
+
 >>(x::Bool, c::UInt) = Int(x) >> c
 <<(x::Bool, c::UInt) = Int(x) << c
 >>>(x::Bool, c::UInt) = Int(x) >>> c
diff --git a/base/exports.jl b/base/exports.jl
index 2a8ed97d4598f..9a6cdce86b38c 100644
--- a/base/exports.jl
+++ b/base/exports.jl
@@ -169,6 +169,10 @@ export
     ≢,
     xor,
     ⊻,
+    nand,
+    nor,
+    ⊼,
+    ⊽,
     %,
     ÷,
     &,
diff --git a/base/gmp.jl b/base/gmp.jl
index d0446e7d3d169..b957ab22a4bea 100644
--- a/base/gmp.jl
+++ b/base/gmp.jl
@@ -4,7 +4,7 @@ module GMP
 
 export BigInt
 
-import .Base: *, +, -, /, <, <<, >>, >>>, <=, ==, >, >=, ^, (~), (&), (|), xor,
+import .Base: *, +, -, /, <, <<, >>, >>>, <=, ==, >, >=, ^, (~), (&), (|), xor, nand, nor,
              binomial, cmp, convert, div, divrem, factorial, cld, fld, gcd, gcdx, lcm, mod,
              ndigits, promote_rule, rem, show, isqrt, string, powermod,
              sum, trailing_zeros, trailing_ones, count_ones, tryparse_internal,
diff --git a/base/operators.jl b/base/operators.jl
index 58f6024f71176..8d7d8180c5cde 100644
--- a/base/operators.jl
+++ b/base/operators.jl
@@ -578,6 +578,8 @@ identity(x) = x
 xor(x::Integer) = x
 
 const ⊻ = xor
+const ⊼ = nand
+const ⊽ = nor
 
 # foldl for argument lists. expand fully up to a point, then
 # switch to a loop. this allows small cases like `a+b+c+d` to be managed
diff --git a/doc/src/base/math.md b/doc/src/base/math.md
index e0f094572a398..8f8a038df83c0 100644
--- a/doc/src/base/math.md
+++ b/doc/src/base/math.md
@@ -49,6 +49,8 @@ Base.:(~)
 Base.:(&)
 Base.:(|)
 Base.xor
+Base.nand
+Base.nor
 Base.:(!)
 &&
 ||
diff --git a/doc/src/manual/mathematical-operations.md b/doc/src/manual/mathematical-operations.md
index 850c1f15bbd17..8df8505ac5cc6 100644
--- a/doc/src/manual/mathematical-operations.md
+++ b/doc/src/manual/mathematical-operations.md
@@ -82,6 +82,8 @@ are supported on all primitive integer types:
 | `x & y`    | bitwise and                                                              |
 | `x \| y`   | bitwise or                                                               |
 | `x ⊻ y`    | bitwise xor (exclusive or)                                               |
+| `x ⊼ y`    | bitwise nand (not and)                                                   |
+| `x ⊽ y`    | bitwise nor (not or)                                                     |
 | `x >>> y`  | [logical shift](https://en.wikipedia.org/wiki/Logical_shift) right       |
 | `x >> y`   | [arithmetic shift](https://en.wikipedia.org/wiki/Arithmetic_shift) right |
 | `x << y`   | logical/arithmetic shift left                                            |
@@ -104,6 +106,18 @@ julia> 123 ⊻ 234
 julia> xor(123, 234)
 145
 
+julia> nand(123, 123)
+-124
+
+julia> 123 ⊼ 123
+-124
+
+julia> nor(123, 124)
+-128
+
+julia> 123 ⊽ 124
+-128
+
 julia> ~UInt32(123)
 0xffffff84
 
diff --git a/stdlib/REPL/src/latex_symbols.jl b/stdlib/REPL/src/latex_symbols.jl
index 57e41ed670538..bfd2e1fbbb015 100644
--- a/stdlib/REPL/src/latex_symbols.jl
+++ b/stdlib/REPL/src/latex_symbols.jl
@@ -110,6 +110,8 @@ const latex_symbols = Dict(
     "\\backpprime" => "‶",
     "\\backppprime" => "‷",
     "\\xor" => "⊻",
+    "\\nand" => "⊼",
+    "\\nor" => "⊽",
     "\\iff" => "⟺",
     "\\implies" => "⟹",
     "\\impliedby" => "⟸",
@@ -2651,4 +2653,6 @@ const symbols_latex_canonical = Dict(
     "→" => "\\to",
     "ε" => "\\varepsilon",
     "⊻" => "\\xor",
+    "⊼" => "\\nand",
+    "⊽" => "\\nor",
 )
diff --git a/test/bitarray.jl b/test/bitarray.jl
index a5591fca26de7..cee7624d9a81a 100644
--- a/test/bitarray.jl
+++ b/test/bitarray.jl
@@ -832,6 +832,8 @@ timesofar("unary arithmetic")
         @check_bit_operation broadcast(&, b1, b2)  BitMatrix
         @check_bit_operation broadcast(|, b1, b2)  BitMatrix
         @check_bit_operation broadcast(xor, b1, b2)  BitMatrix
+        @check_bit_operation broadcast(nand, b1, b2)  BitMatrix
+        @check_bit_operation broadcast(nor, b1, b2)  BitMatrix
         @check_bit_operation (+)(b1, b2)  Matrix{Int}
         @check_bit_operation (-)(b1, b2)  Matrix{Int}
         @check_bit_operation broadcast(*, b1, b2) BitMatrix
@@ -861,6 +863,8 @@ timesofar("unary arithmetic")
         @check_bit_operation broadcast(&, b0, b0)  BitVector
         @check_bit_operation broadcast(|, b0, b0)  BitVector
         @check_bit_operation broadcast(xor, b0, b0)  BitVector
+        @check_bit_operation broadcast(nand, b0, b0)  BitVector
+        @check_bit_operation broadcast(nor, b0, b0)  BitVector
         @check_bit_operation broadcast(*, b0, b0) BitVector
         @check_bit_operation (*)(b0, b0') BitMatrix
     end
@@ -871,6 +875,8 @@ timesofar("unary arithmetic")
         @check_bit_operation broadcast(&, b1, i2)  Matrix{Int}
         @check_bit_operation broadcast(|, b1, i2)  Matrix{Int}
         @check_bit_operation broadcast(xor, b1, i2)  Matrix{Int}
+        @check_bit_operation broadcast(nand, b1, i2)  Matrix{Int}
+        @check_bit_operation broadcast(nor, b1, i2)  Matrix{Int}
         @check_bit_operation (+)(b1, i2)  Matrix{Int}
         @check_bit_operation (-)(b1, i2)  Matrix{Int}
         @check_bit_operation broadcast(*, b1, i2) Matrix{Int}
@@ -902,6 +908,8 @@ timesofar("unary arithmetic")
         @check_bit_operation broadcast(&, i1, b2)  Matrix{Int}
         @check_bit_operation broadcast(|, i1, b2)  Matrix{Int}
         @check_bit_operation broadcast(xor, i1, b2)  Matrix{Int}
+        @check_bit_operation broadcast(nand, i1, b2)  Matrix{Int}
+        @check_bit_operation broadcast(nor, i1, b2)  Matrix{Int}
         @check_bit_operation broadcast(+, i1, b2)  Matrix{Int}
         @check_bit_operation broadcast(-, i1, b2)  Matrix{Int}
         @check_bit_operation broadcast(*, i1, b2) Matrix{Int}
@@ -909,6 +917,8 @@ timesofar("unary arithmetic")
         @check_bit_operation broadcast(&, u1, b2)  Matrix{UInt8}
         @check_bit_operation broadcast(|, u1, b2)  Matrix{UInt8}
         @check_bit_operation broadcast(xor, u1, b2)  Matrix{UInt8}
+        @check_bit_operation broadcast(nand, u1, b2)  Matrix{UInt8}
+        @check_bit_operation broadcast(nor, u1, b2)  Matrix{UInt8}
         @check_bit_operation broadcast(+, u1, b2)  Matrix{UInt8}
         @check_bit_operation broadcast(-, u1, b2)  Matrix{UInt8}
         @check_bit_operation broadcast(*, u1, b2) Matrix{UInt8}
@@ -986,6 +996,14 @@ timesofar("unary arithmetic")
         @check_bit_operation broadcast(xor, b1, false)  BitMatrix
         @check_bit_operation broadcast(xor, true, b1)   BitMatrix
         @check_bit_operation broadcast(xor, false, b1)  BitMatrix
+        @check_bit_operation broadcast(nand, b1, true)   BitMatrix
+        @check_bit_operation broadcast(nand, b1, false)  BitMatrix
+        @check_bit_operation broadcast(nand, true, b1)   BitMatrix
+        @check_bit_operation broadcast(nand, false, b1)  BitMatrix
+        @check_bit_operation broadcast(nor, b1, true)   BitMatrix
+        @check_bit_operation broadcast(nor, b1, false)  BitMatrix
+        @check_bit_operation broadcast(nor, true, b1)   BitMatrix
+        @check_bit_operation broadcast(nor, false, b1)  BitMatrix
         @check_bit_operation broadcast(+, b1, true)   Matrix{Int}
         @check_bit_operation broadcast(+, b1, false)  Matrix{Int}
         @check_bit_operation broadcast(-, b1, true)   Matrix{Int}
@@ -1002,12 +1020,18 @@ timesofar("unary arithmetic")
         @check_bit_operation broadcast(&, b1, b2)  BitMatrix
         @check_bit_operation broadcast(|, b1, b2)  BitMatrix
         @check_bit_operation broadcast(xor, b1, b2)  BitMatrix
+        @check_bit_operation broadcast(nand, b1, b2)  BitMatrix
+        @check_bit_operation broadcast(nor, b1, b2)  BitMatrix
         @check_bit_operation broadcast(&, b2, b1)  BitMatrix
         @check_bit_operation broadcast(|, b2, b1)  BitMatrix
         @check_bit_operation broadcast(xor, b2, b1)  BitMatrix
+        @check_bit_operation broadcast(nand, b2, b1)  BitMatrix
+        @check_bit_operation broadcast(nor, b2, b1)  BitMatrix
         @check_bit_operation broadcast(&, b1, i2)  Matrix{Int}
         @check_bit_operation broadcast(|, b1, i2)  Matrix{Int}
         @check_bit_operation broadcast(xor, b1, i2)  Matrix{Int}
+        @check_bit_operation broadcast(nand, b1, i2)  Matrix{Int}
+        @check_bit_operation broadcast(nor, b1, i2)  Matrix{Int}
         @check_bit_operation broadcast(+, b1, i2)  Matrix{Int}
         @check_bit_operation broadcast(-, b1, i2)  Matrix{Int}
         @check_bit_operation broadcast(*, b1, i2) Matrix{Int}
@@ -1018,6 +1042,8 @@ timesofar("unary arithmetic")
         @check_bit_operation broadcast(&, b1, u2)  Matrix{UInt8}
         @check_bit_operation broadcast(|, b1, u2)  Matrix{UInt8}
         @check_bit_operation broadcast(xor, b1, u2)  Matrix{UInt8}
+        @check_bit_operation broadcast(nand, b1, u2)  Matrix{UInt8}
+        @check_bit_operation broadcast(nor, b1, u2)  Matrix{UInt8}
         @check_bit_operation broadcast(+, b1, u2)  Matrix{UInt8}
         @check_bit_operation broadcast(-, b1, u2)  Matrix{UInt8}
         @check_bit_operation broadcast(*, b1, u2) Matrix{UInt8}
@@ -1086,6 +1112,14 @@ timesofar("unary arithmetic")
         @check_bit_operation broadcast(xor, b1, transpose(b3))  BitMatrix
         @check_bit_operation broadcast(xor, b2, b1)             BitMatrix
         @check_bit_operation broadcast(xor, transpose(b3), b1)  BitMatrix
+        @check_bit_operation broadcast(nand, b1, b2)             BitMatrix
+        @check_bit_operation broadcast(nand, b1, transpose(b3))  BitMatrix
+        @check_bit_operation broadcast(nand, b2, b1)             BitMatrix
+        @check_bit_operation broadcast(nand, transpose(b3), b1)  BitMatrix
+        @check_bit_operation broadcast(nor, b1, b2)             BitMatrix
+        @check_bit_operation broadcast(nor, b1, transpose(b3))  BitMatrix
+        @check_bit_operation broadcast(nor, b2, b1)             BitMatrix
+        @check_bit_operation broadcast(nor, transpose(b3), b1)  BitMatrix
         @check_bit_operation broadcast(+, b1, b2)             Matrix{Int}
         @check_bit_operation broadcast(+, b1, transpose(b3))  Matrix{Int}
         @check_bit_operation broadcast(+, b2, b1)             Matrix{Int}
@@ -1390,6 +1424,8 @@ timesofar("reductions")
         @test map(&, b1, b2) == map((x,y)->x&y, b1, b2) == broadcast(&, b1, b2)
         @test map(|, b1, b2) == map((x,y)->x|y, b1, b2) == broadcast(|, b1, b2)
         @test map(⊻, b1, b2) == map((x,y)->x⊻y, b1, b2) == broadcast(⊻, b1, b2) == broadcast(xor, b1, b2)
+        @test map(⊼, b1, b2) == map((x,y)->x⊼y, b1, b2) == broadcast(⊼, b1, b2) == broadcast(nand, b1, b2)
+        @test map(⊽, b1, b2) == map((x,y)->x⊽y, b1, b2) == broadcast(⊽, b1, b2) == broadcast(nor, b1, b2)
 
         @test map(^, b1, b2) == map((x,y)->x^y, b1, b2) == b1 .^ b2
         @test map(*, b1, b2) == map((x,y)->x*y, b1, b2) == b1 .* b2
diff --git a/test/gmp.jl b/test/gmp.jl
index 96ffdedb7a93c..3d7317b37bacc 100644
--- a/test/gmp.jl
+++ b/test/gmp.jl
@@ -241,6 +241,12 @@ end
     @test xor(a, b, c, d, f) == parse(BigInt,"-2413804710837418037418307081437316711364709261074607933698")
     @test xor(a, b, c, d, f, g) == parse(BigInt,"2413804710837418037418307081437316711364709261074607933697")
 
+    @test nand(a, b) == parse(BigInt,"-125")
+    @test ⊼(a, b) == parse(BigInt,"-125")
+
+    @test nor(a, b) == parse(BigInt,"-327424")
+    @test ⊽(a, b) == parse(BigInt,"-327424")
+
     @test (&)(a, b) == parse(BigInt,"124")
     @test (&)(a, b, c) == parse(BigInt,"72")
     @test (&)(a, b, c, d) == parse(BigInt,"8")
diff --git a/test/missing.jl b/test/missing.jl
index 4c704c4ce64f8..c20395b70b564 100644
--- a/test/missing.jl
+++ b/test/missing.jl
@@ -83,7 +83,7 @@ end
     arithmetic_operators = [+, -, *, /, ^, Base.div, Base.mod, Base.fld, Base.rem]
 
     # All unary operators return missing when evaluating missing
-    for f in [!, ~, +, -, *, &, |, xor]
+    for f in [!, ~, +, -, *, &, |, xor, nand, nor]
         @test ismissing(f(missing))
     end
 
@@ -128,6 +128,22 @@ end
     @test ismissing(xor(true, missing))
     @test ismissing(xor(missing, false))
     @test ismissing(xor(false, missing))
+    @test ismissing(nand(missing, true))
+    @test ismissing(nand(true, missing))
+    @test nand(missing, false) == true
+    @test nand(false, missing) == true
+    @test ismissing(⊼(missing, true))
+    @test ismissing(⊼(true, missing))
+    @test ⊼(missing, false) == true
+    @test ⊼(false, missing) == true
+    @test nor(missing, true) == false
+    @test nor(true, missing) == false
+    @test ismissing(nor(missing, false))
+    @test ismissing(nor(false, missing))
+    @test ⊽(missing, true) == false
+    @test ⊽(true, missing) == false
+    @test ismissing(⊽(missing, false))
+    @test ismissing(⊽(false, missing))
 
     @test ismissing(missing & 1)
     @test ismissing(1 & missing)
@@ -135,6 +151,14 @@ end
     @test ismissing(1 | missing)
     @test ismissing(xor(missing, 1))
     @test ismissing(xor(1, missing))
+    @test ismissing(nand(missing, 1))
+    @test ismissing(nand(1, missing))
+    @test ismissing(⊼(missing, 1))
+    @test ismissing(⊼(1, missing))
+    @test ismissing(nor(missing, 1))
+    @test ismissing(nor(1, missing))
+    @test ismissing(⊽(missing, 1))
+    @test ismissing(⊽(1, missing))
 end
 
 @testset "* string concatenation" begin
diff --git a/test/numbers.jl b/test/numbers.jl
index a76f5726492df..e62e1a798221d 100644
--- a/test/numbers.jl
+++ b/test/numbers.jl
@@ -38,6 +38,24 @@ const ≣ = isequal # convenient for comparing NaNs
     @test xor(true,  false) == true
     @test xor(false, true)  == true
     @test xor(true,  true)  == false
+
+    @test false ⊼ false == true
+    @test true ⊼ false == true
+    @test false ⊼ true == true
+    @test true ⊼ true == false
+    @test nand(false, false) == true
+    @test nand(true, false) == true
+    @test nand(false, true) == true
+    @test nand(true, true) == false
+
+    @test false ⊽ false == true
+    @test true ⊽ false == false
+    @test false ⊽ true == false
+    @test true ⊽ true == false
+    @test nor(false, false) == true
+    @test nor(true, false) == false
+    @test nor(false, true) == false
+    @test nor(true, true) == false
 end
 @testset "bool operator" begin
     @test Bool(false) == false