diff --git a/docs/src/manual/functions.md b/docs/src/manual/functions.md
index 9e0b8e673..4238f84f1 100644
--- a/docs/src/manual/functions.md
+++ b/docs/src/manual/functions.md
@@ -188,3 +188,21 @@ function Symbolics.derivative(interp::AbstractInterpolation, args::NTuple{1, Any
     Symbolics.unwrap(derivative(interp, Symbolics.wrap(args[1])))
 end
 ```
+
+## Inverse function registration
+
+Symbolics.jl allows defining and querying the inverses of functions.
+
+```@docs
+inverse
+left_inverse
+right_inverse
+@register_inverse
+has_inverse
+has_left_inverse
+has_right_inverse
+```
+
+Symbolics.jl implements inverses for standard trigonometric and logarithmic functions,
+as well as their variants from `NaNMath`. It also implements inverses of
+`ComposedFunction`s.
diff --git a/src/Symbolics.jl b/src/Symbolics.jl
index 386ddd88d..cd84fd16a 100644
--- a/src/Symbolics.jl
+++ b/src/Symbolics.jl
@@ -19,11 +19,9 @@ using Primes
 
 using Reexport
 
-using DomainSets
-
 using Setfield
 
-import DomainSets: Domain
+import DomainSets: Domain, DomainSets
 
 using TermInterface
 import TermInterface: maketerm, iscall, operation, arguments, metadata
@@ -234,4 +232,7 @@ function __init__()
     end
 end
 
+export inverse, left_inverse, right_inverse, @register_inverse, has_inverse, has_left_inverse, has_right_inverse
+include("inverse.jl")
+
 end # module
diff --git a/src/inverse.jl b/src/inverse.jl
new file mode 100644
index 000000000..0d541d5c2
--- /dev/null
+++ b/src/inverse.jl
@@ -0,0 +1,176 @@
+"""
+    inverse(f)
+
+Given a single-input single-output function `f`, return its inverse `g`. This requires
+that `f` is bijective. If `inverse` is defined for a function, `left_inverse` and
+`right_inverse` should return `inverse(f)`. `inverse(g)` should also be defined to
+return `f`.
+
+See also: [`left_inverse`](@ref), [`right_inverse`](@ref), [`@register_inverse`](@ref).
+"""
+function inverse end
+
+"""
+    left_inverse(f)
+
+Given a single-input single-output function `f`, return its left inverse `g`. This
+requires that `f` is injective. If `left_inverse` is defined for a function,
+`right_inverse` and `inverse` must not be defined and should error. `right_inverse(g)`
+should also be defined to return `f`.
+
+See also: [`inverse`](@ref), [`right_inverse`](@ref), [`@register_inverse`](@ref).
+"""
+function left_inverse end
+
+"""
+    right_inverse(f)
+
+Given a single-input single-output function `f`, return its right inverse `g`. This
+requires that `f` is surjective. If `right_inverse` is defined for a function,
+`left_inverse` and `inverse` must not be defined and should error. `left_inverse(g)`
+should also be defined to return `f`.
+
+See also [`inverse`](@ref), [`left_inverse`](@ref), [`@register_inverse`](@ref).
+"""
+function right_inverse end
+
+"""
+    @register_inverse f g
+    @register_inverse f g left
+    @register_inverse f g right
+
+Mark `f` and `g` as inverses of each other. By default, assume that `f` and `g` are
+bijective. Also defines `left_inverse` and `right_inverse` to call `inverse`. If the
+third argument is `left`, assume that `f` is injective and `g` is its left inverse. If
+the third argument is `right`, assume that `f` is surjective and `g` is its right
+inverse.
+"""
+macro register_inverse(f, g, dir::QuoteNode = :(:both))
+    dir = dir.value
+    if dir == :both
+        quote
+            (::typeof($inverse))(::typeof($f)) = $g
+            (::typeof($inverse))(::typeof($g)) = $f
+            (::typeof($left_inverse))(::typeof($f)) = $(inverse)($f)
+            (::typeof($right_inverse))(::typeof($f)) = $(inverse)($f)
+            (::typeof($left_inverse))(::typeof($g)) = $(inverse)($g)
+            (::typeof($right_inverse))(::typeof($g)) = $(inverse)($g)
+        end
+    elseif dir == :left
+        quote
+            (::typeof($left_inverse))(::typeof($f)) = $g
+            (::typeof($right_inverse))(::typeof($g)) = $f
+        end
+    elseif dir == :right
+        quote
+            (::typeof($right_inverse))(::typeof($f)) = $g
+            (::typeof($left_inverse))(::typeof($g)) = $f
+        end
+    else
+        throw(ArgumentError("The third argument to `@register_inverse` must be `left` or `right`"))
+    end
+end
+
+"""
+    $(TYPEDSIGNATURES)
+
+Check if the provided function has an inverse defined via [`inverse`](@ref). Uses
+`hasmethod` to perform the check.
+"""
+has_inverse(::T) where {T} = hasmethod(inverse, Tuple{T})
+
+"""
+    $(TYPEDSIGNATURES)
+
+Check if the provided function has a left inverse defined via [`left_inverse`](@ref)
+Uses `hasmethod` to perform the check.
+"""
+has_left_inverse(::T) where {T} = hasmethod(left_inverse, Tuple{T})
+
+"""
+    $(TYPEDSIGNATURES)
+
+Check if the provided function has a left inverse defined via [`left_inverse`](@ref)
+Uses `hasmethod` to perform the check.
+"""
+has_right_inverse(::T) where {T} = hasmethod(right_inverse, Tuple{T})
+
+"""
+    $(TYPEDSIGNATURES)
+
+A simple utility function which returns the square of the input. Used to define
+the inverse of `sqrt`.
+"""
+square(x) = x ^ 2
+
+"""
+    $(TYPEDSIGNATURES)
+
+A simple utility function which returns the cube of the input. Used to define
+the inverse of `cbrt`.
+"""
+cube(x) = x ^ 3
+
+"""
+    $(TYPEDSIGNATURES)
+
+A simple utility function which takes `x` and returns `acos(x) / pi`. Used to
+define the inverse of `acospi`.
+"""
+acosbypi(x) = acos(x) / pi
+
+@register_inverse sin asin
+@register_inverse cos acos
+@register_inverse tan atan
+@register_inverse csc acsc
+@register_inverse sec asec
+@register_inverse cot acot
+@register_inverse sind asind
+@register_inverse cosd acosd
+@register_inverse tand atand
+@register_inverse cscd acscd
+@register_inverse secd asecd
+@register_inverse cotd acotd
+@register_inverse sinh asinh
+@register_inverse cosh acosh
+@register_inverse tanh atanh
+@register_inverse csch acsch
+@register_inverse sech asech
+@register_inverse coth acoth
+@register_inverse cospi acosbypi
+@register_inverse SpecialFunctions.digamma SpecialFunctions.invdigamma
+@register_inverse log exp
+@register_inverse log2 exp2
+@register_inverse log10 exp10
+@register_inverse log1p expm1
+@register_inverse deg2rad rad2deg
+@register_inverse sqrt square :left
+@register_inverse cbrt cube
+@register_inverse NaNMath.sin NaNMath.asin
+@register_inverse NaNMath.cos NaNMath.acos
+# can't use macro since it would be a re-definition of `inverse(atan)`
+inverse(::typeof(NaNMath.tan)) = inverse(tan)
+inverse(::typeof(NaNMath.acosh)) = inverse(acosh)
+inverse(::typeof(NaNMath.atanh)) = inverse(atanh)
+inverse(::typeof(NaNMath.log)) = inverse(log)
+inverse(::typeof(NaNMath.log10)) = inverse(log10)
+inverse(::typeof(NaNMath.log1p)) = inverse(log1p)
+inverse(::typeof(NaNMath.log2)) = inverse(log2)
+left_inverse(::typeof(NaNMath.sqrt)) = left_inverse(sqrt)
+
+function inverse(f::ComposedFunction)
+    return inverse(f.inner) ∘ inverse(f.outer)
+end
+has_inverse(f::ComposedFunction) = has_inverse(f.inner) && has_inverse(f.outer)
+function left_inverse(f::ComposedFunction)
+    return left_inverse(f.inner) ∘ left_inverse(f.outer)
+end
+function has_left_inverse(f::ComposedFunction)
+    return has_left_inverse(f.inner) && has_left_inverse(f.outer)
+end
+function right_inverse(f::ComposedFunction)
+    return right_inverse(f.inner) ∘ right_inverse(f.outer)
+end
+function has_right_inverse(f::ComposedFunction)
+    return has_right_inverse(f.inner) && has_right_inverse(f.outer)
+end
diff --git a/test/inverse.jl b/test/inverse.jl
new file mode 100644
index 000000000..091fa458a
--- /dev/null
+++ b/test/inverse.jl
@@ -0,0 +1,18 @@
+using Symbolics
+
+@test inverse(sin) == left_inverse(sin) == right_inverse(sin) == asin
+@test inverse(asin) == left_inverse(asin) == right_inverse(asin) == sin
+@test has_inverse(sin) && has_left_inverse(sin) && has_right_inverse(sin)
+fn = left_inverse(sqrt)
+@test right_inverse(fn) == sqrt
+@test_throws MethodError inverse(sqrt)
+@test_throws MethodError right_inverse(sqrt)
+@test !has_inverse(fn) && !has_left_inverse(fn)
+@test !has_inverse(sqrt) && !has_right_inverse(sqrt)
+@test has_inverse(sin ∘ cos)
+@test !has_inverse(sin ∘ sqrt)
+@test has_left_inverse(sin ∘ sqrt)
+@test inverse(sin ∘ cos) == acos ∘ asin
+@test inverse(inverse(sin ∘ cos)) == sin ∘ cos
+@test right_inverse(left_inverse(sin ∘ sqrt)) == sin ∘ sqrt
+@test inverse(sin ∘ cos ∘ tan) == atan ∘ (acos ∘ asin)
diff --git a/test/runtests.jl b/test/runtests.jl
index 9c19ec9fc..8db2d5c9d 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -60,6 +60,7 @@ if GROUP == "All" || GROUP == "Core"
         @safetestset "Show Test" begin include("show.jl") end
         @safetestset "Utility Function Test" begin include("utils.jl") end
         @safetestset "RootFinding solver" begin include("solver.jl") end
+        @safetestset "Function inverses test" begin include("inverse.jl") end
     end
 end