diff --git a/docs/src/operations.md b/docs/src/operations.md
index 64e9779..6f7a8ec 100644
--- a/docs/src/operations.md
+++ b/docs/src/operations.md
@@ -89,3 +89,19 @@ using Bijections # hide
 b = Bijection(1 => "alpha", 2 => "beta", 3 => "gamma");
 for (x, y) in b; println("$x --> $y"); end
 ```
+
+## Composition
+
+Given two `Bijection`s `a` and `b`, their composition `c = a ∘ b` or `c = compose(a, b)` is a new `Bijection` with the property that `c[x] = a[b[x]]` for all `x` in the
+domain of `b`.
+
+```jldoctest
+julia> a = Bijection{Int,Int}(1 => 10, 2 => 20);
+
+julia> b = Bijection{String,Int}("hi" => 1, "bye" => 2);
+
+julia> c = a ∘ b;
+
+julia> c["hi"]
+10
+```
diff --git a/src/Bijections.jl b/src/Bijections.jl
index 020bae5..5710fcf 100644
--- a/src/Bijections.jl
+++ b/src/Bijections.jl
@@ -2,7 +2,7 @@ module Bijections
 
 using Serialization: Serialization
 
-export Bijection, active_inv, inverse, hasvalue
+export Bijection, active_inv, inverse, hasvalue, compose
 
 struct Bijection{K,V,F,Finv} <: AbstractDict{K,V}
     f::F          # map from domain to range
@@ -269,4 +269,58 @@ function Serialization.deserialize(
     return B(f)
 end
 
+# WARN this uses internals so it's dangerous!
+"""
+    C = composed_dict_type(A::Type{<:AbstractDict}, B::Type{<:AbstractDict})
+
+Returns the type of the forward dictionary of `(a ∘ b)` where `A` and `B` are
+the types of the forward-dictionaries of `a` and `b`, respectively.
+
+For any combination of a `IdDict` and a `Dict`, the result will be an `IdDict`.
+Otherwise, return `A` with the types of keys and values adjusted so that the
+resulting dict maps keys of `b` to values of `a`.
+"""
+function composed_dict_type(
+    A::Type{<:AbstractDict{AK,AV}}, ::Type{<:AbstractDict{BK,BV}}
+) where {AK,AV,BK,BV}
+    return A.name.wrapper{BK,AV}
+end
+function composed_dict_type(::Type{Dict{AK,AV}}, ::Type{Dict{BK,BV}}) where {AK,AV,BK,BV}
+    Dict{BK,AV}
+end
+function composed_dict_type(::Type{Dict{AK,AV}}, ::Type{IdDict{BK,BV}}) where {AK,AV,BK,BV}
+    IdDict{BK,AV}
+end
+function composed_dict_type(::Type{IdDict{AK,AV}}, ::Type{Dict{BK,BV}}) where {AK,AV,BK,BV}
+    IdDict{BK,AV}
+end
+function composed_dict_type(
+    ::Type{IdDict{AK,AV}}, ::Type{IdDict{BK,BV}}
+) where {AK,AV,BK,BV}
+    IdDict{BK,AV}
+end
+
+"""
+    c = (∘)(a::Bijection, b::Bijection)
+    c = compose(a, b)
+
+The result of `a ∘ b` or `compose(a, b)` is a new `Bijection` `c` such that
+`c[x]` is `a[b[x]]` for `x` in the domain of `b`. The internal type of the
+    forward mapping is determined by [`composed_dict_type`](@ref), and the type
+    of the backward mapping is determined by [`inverse_dict_type`](@ref).
+"""
+function compose(
+    a::Bijection{AK,AV,AF,AFinv}, b::Bijection{BK,BV,BF,BFinv}
+) where {AK,AV,AF,AFinv,BK,BV,BF,BFinv}
+    CF = composed_dict_type(AF, BF)
+    CFinv = inverse_dict_type(CF)
+    c = Bijection{BK,AV,CF,CFinv}()
+    for x in keys(b)
+        c[x] = a[b[x]]
+    end
+    return c
+end
+
+Base.:(∘)(a::Bijection, b::Bijection) = compose(a, b)
+
 end # end of module Bijections
diff --git a/test/runtests.jl b/test/runtests.jl
index 495a02a..3d033f2 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -38,6 +38,113 @@ using Serialization
     @test Bijection(collect(b)) == b
 end
 
+# check composition
+@testset "Composition" begin
+    a = Bijection{Int,Float64}()
+    a[1] = 10.0
+    a[2] = 20.0
+
+    b = Bijection{String,Int}()
+    b["hi"] = 1
+    b["bye"] = 2
+
+    c = a ∘ b
+    @test c["hi"] == 10.0
+    @test c(10.0) == "hi"
+
+    @test compose(a, b) == c
+
+    # Mutable objects (1)
+    # - mutable keys/values in `a`, non-mutable keys/values in `b`
+
+    a = Bijection{
+        Int64,Vector{Int64},IdDict{Int64,Vector{Int64}},IdDict{Vector{Int64},Int64}
+    }()
+    A₁ = [1, 2, 3]
+    A₂ = [3, 4, 5]
+    a[1] = A₁
+    a[2] = A₂
+
+    c = a ∘ b
+    @test c isa Bijection{
+        String,Vector{Int64},IdDict{String,Vector{Int64}},IdDict{Vector{Int64},String}
+    }
+    @test c["hi"] === A₁
+    @test c(A₁) == "hi"
+    A₁[1] = 10
+    @test c["hi"] === A₁
+    @test c(A₁) == "hi"
+    @test_throws KeyError c([1, 2, 3])
+
+    # Mutable objects (2)
+    # - Mutable keys/values both in `a` and `b`
+
+    a = Bijection{
+        Int64,Vector{Int64},IdDict{Int64,Vector{Int64}},IdDict{Vector{Int64},Int64}
+    }()
+    A₁ = [1, 2, 3]
+    A₂ = [3, 4, 5]
+    a[1] = A₁
+    a[2] = A₂
+
+    b = Bijection{
+        Vector{Int64},Int64,IdDict{Vector{Int64},Int64},IdDict{Int64,Vector{Int64}}
+    }()
+    b₁ = [1, 2, 3]
+    b₂ = [3, 4, 5]
+    @test b₁ ≢ A₁
+    @test b₂ ≢ A₂
+    b[b₁] = 1
+    b[b₂] = 2
+
+    c = a ∘ b
+    @test c[b₁] === A₁
+    @test c(A₁) === b₁
+
+    b₁[1] = 10
+    @test c[b₁] === A₁
+
+    # Mutable objects (3)
+    # - Non-mutable keys/values in `a`, mutable keys/values in `b`
+
+    a = Bijection{Int,Float64}()
+    a[1] = 10.0
+    a[2] = 20.0
+
+    b = Bijection{
+        Vector{Int64},Int64,IdDict{Vector{Int64},Int64},Dict{Int64,Vector{Int64}}
+    }()
+    b₁ = [1, 2, 3]
+    b₂ = [3, 4, 5]
+    b[b₁] = 1
+    b[b₂] = 2
+
+    c = a ∘ b
+    @test c isa Bijection{
+        Vector{Int64},Float64,IdDict{Vector{Int64},Float64},IdDict{Float64,Vector{Int64}}
+    }
+    @test c[b₁] == 10.0
+    @test c(10.0) === b₁
+
+    b₁[1] = 10
+    @test c[b₁] == 10.0
+
+    # ImmutableDict (testing the fallback of composed_dict_type)
+
+    a = Bijection{Int,Float64}()
+    a[1] = 10.0
+    a[2] = 20.0
+
+    b = Bijection(
+        Base.ImmutableDict("hi" => 1, "bye" => 2), Base.ImmutableDict(1 => "hi", 2 => "bye")
+    )
+
+    c = a ∘ b
+    @test c isa Bijection{String,Float64,Dict{String,Float64},Dict{Float64,String}}
+    @test c["hi"] == 10.0
+    @test c(10.0) == "hi"
+end
+
 # Test empty constructor
 @testset "empty_constructor" begin
     b = Bijection{Int,String}()