diff --git a/.github/workflows/ci.yml b/.github/workflows/ci.yml
index bfec232..2a4cf15 100644
--- a/.github/workflows/ci.yml
+++ b/.github/workflows/ci.yml
@@ -17,8 +17,9 @@ jobs:
       fail-fast: false
       matrix:
         version:
-          - '1.5' # Replace this with the minimum Julia version that your package supports.
-          # - '1'   # automatically expands to the latest stable 1.x release of Julia
+          - '1.0'
+          - '1.6' # Replace this with the minimum Julia version that your package supports.
+          - '1'   # automatically expands to the latest stable 1.x release of Julia
           - 'nightly'
         os:
           - ubuntu-latest
diff --git a/Project.toml b/Project.toml
index 1f8b8a9..84dc030 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,11 +1,11 @@
 name = "Functors"
 uuid = "d9f16b24-f501-4c13-a1f2-28368ffc5196"
 authors = ["Mike J Innes <mike.j.innes@gmail.com>"]
-version = "0.2.7"
+version = "0.2.8"
 
 [compat]
-julia = "1"
 Documenter = "0.27"
+julia = "1"
 
 [extras]
 Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4"
diff --git a/src/Functors.jl b/src/Functors.jl
index dcf21f8..5247df1 100644
--- a/src/Functors.jl
+++ b/src/Functors.jl
@@ -23,6 +23,7 @@ usually using the macro [@functor](@ref).
 """
 functor
 
+@static if VERSION >= v"1.5"  # var"@functor" doesn't work on 1.0, temporarily disable
 """
     @functor T
     @functor T (x,)
@@ -65,6 +66,7 @@ TwoThirds(Foo(10, 20), Foo(3, 4), 560)
 ```
 """
 var"@functor"
+end  # VERSION
 
 """
     Functors.isleaf(x)
@@ -182,6 +184,16 @@ This function walks (maps) over `xs` calling the continuation `f'` to continue t
 julia> fmap(x -> 10x, m, walk=(f, x) -> x isa Bar ? x : Functors._default_walk(f, x))
 Foo(Bar([1, 2, 3]), (40, 50, Bar(Foo(6, 7))))
 ```
+
+The behaviour when the same node appears twice can be altered by giving a value
+to the `prune` keyword, which is then used in place of all but the first:
+
+```jldoctest
+julia> twice = [1, 2];
+
+julia> fmap(float, (x = twice, y = [1,2], z = twice); prune = missing)
+(x = [1.0, 2.0], y = [1.0, 2.0], z = missing)
+```
 """
 fmap
 
diff --git a/src/functor.jl b/src/functor.jl
index 608dbd3..b8cb8f5 100644
--- a/src/functor.jl
+++ b/src/functor.jl
@@ -3,7 +3,7 @@ functor(T, x) = (), _ -> x
 functor(x) = functor(typeof(x), x)
 
 functor(::Type{<:Tuple}, x) = x, y -> y
-functor(::Type{<:NamedTuple}, x) = x, y -> y
+functor(::Type{<:NamedTuple{L}}, x) where L = NamedTuple{L}(map(s -> getproperty(x, s), L)), identity
 
 functor(::Type{<:AbstractArray}, x) = x, y -> y
 functor(::Type{<:AbstractArray{<:Number}}, x) = (), _ -> x
@@ -43,12 +43,11 @@ function _default_walk(f, x)
   re(map(f, func))
 end
 
-function fmap(f, x; exclude = isleaf, walk = _default_walk, cache = IdDict())
-  haskey(cache, x) && return cache[x]
-  y = exclude(x) ? f(x) : walk(x -> fmap(f, x, exclude = exclude, walk = walk, cache = cache), x)
-  cache[x] = y
+struct NoKeyword end
 
-  return y
+function fmap(f, x; exclude = isleaf, walk = _default_walk, cache = IdDict(), prune = NoKeyword())
+  haskey(cache, x) && return prune isa NoKeyword ? cache[x] : prune
+  cache[x] = exclude(x) ? f(x) : walk(x -> fmap(f, x; exclude=exclude, walk=walk, cache=cache, prune=prune), x)
 end
 
 ###
@@ -74,27 +73,16 @@ end
 ### Vararg forms
 ###
 
-function fmap(f, x, dx...; cache = IdDict())
-  haskey(cache, x) && return cache[x]
-  cache[x] = isleaf(x) ? f(x, dx...) : _default_walk((x...) -> fmap(f, x..., cache = cache), x, dx...)
+function fmap(f, x, ys...; exclude = isleaf, walk = _default_walk, cache = IdDict(), prune = NoKeyword())
+  haskey(cache, x) && return prune isa NoKeyword ? cache[x] : prune
+  cache[x] = exclude(x) ? f(x, ys...) : walk((xy...,) -> fmap(f, xy...; exclude=exclude, walk=walk, cache=cache, prune=prune), x, ys...)
 end
 
-function functor_tuple(f, x::Tuple, dx::Tuple)
-  map(x, dx) do x, x̄
-    _default_walk(f, x, x̄)
-  end
-end
-functor_tuple(f, x, dx) = f(x, dx)
-functor_tuple(f, x, ::Nothing) = x
-
-function _default_walk(f, x, dx)
+function _default_walk(f, x, ys...)
   func, re = functor(x)
-  map(func, dx) do x, x̄
-    # functor_tuple(f, x, x̄)
-    f(x, x̄)
-  end |> re
+  yfuncs = map(y -> functor(typeof(x), y)[1], ys)
+  re(map(f, func, yfuncs...))
 end
-_default_walk(f, ::Nothing, ::Nothing) = nothing
 
 ###
 ### FlexibleFunctors.jl
@@ -112,9 +100,7 @@ function makeflexiblefunctor(m::Module, T, pfield)
       func = NamedTuple{pfields}(map(p -> getproperty(x, p), pfields))
       return func, re
     end
-
   end
-
 end
 
 function flexiblefunctorm(T, pfield = :params)
diff --git a/test/basics.jl b/test/basics.jl
index 21aa445..914a5cd 100644
--- a/test/basics.jl
+++ b/test/basics.jl
@@ -1,25 +1,16 @@
-struct Foo
-  x
-  y
-end
+
+using Functors: functor
+
+struct Foo; x; y; end
 @functor Foo
 
-struct Bar
-  x
-end
+struct Bar; x; end
 @functor Bar
 
-struct Baz
-  x
-  y
-  z
-end
-@functor Baz (y,)
+struct OneChild3; x; y; z; end
+@functor OneChild3 (y,)
 
-struct NoChildren 
-  x
-  y
-end
+struct NoChildren2; x; y; end
 
 @static if VERSION >= v"1.6"
   @testset "ComposedFunction" begin
@@ -31,6 +22,10 @@ end
   end
 end
 
+###
+### Basic functionality
+###
+
 @testset "Nested" begin
   model = Bar(Foo(1, [1, 2, 3]))
 
@@ -53,6 +48,73 @@ end
   @test fmap(f, x; exclude = x -> x isa AbstractArray) == x
 end
 
+@testset "Property list" begin
+  model = OneChild3(1, 2, 3)
+  model′ = fmap(x -> 2x, model)
+  
+  @test (model′.x, model′.y, model′.z) == (1, 4, 3)
+end
+
+@testset "cache" begin
+  shared = [1,2,3]
+  m1 = Foo(shared, Foo([1,2,3], Foo(shared, [1,2,3])))
+  m1f = fmap(float, m1)
+  @test m1f.x === m1f.y.y.x
+  @test m1f.x !== m1f.y.x
+  m1p = fmapstructure(identity, m1; prune = nothing)
+  @test m1p == (x = [1, 2, 3], y = (x = [1, 2, 3], y = (x = nothing, y = [1, 2, 3])))
+
+  # A non-leaf node can also be repeated:
+  m2 = Foo(Foo(shared, 4), Foo(shared, 4))
+  @test m2.x === m2.y
+  m2f = fmap(float, m2)
+  @test m2f.x.x === m2f.y.x
+  m2p = fmapstructure(identity, m2; prune = Bar(0))
+  @test m2p == (x = (x = [1, 2, 3], y = 4), y = Bar(0))
+
+  # Repeated isbits types should not automatically be regarded as shared:
+  m3 = Foo(Foo(shared, 1:3), Foo(1:3, shared))
+  m3p = fmapstructure(identity, m3; prune = 0)
+  @test m3p.y.y == 0
+  @test_broken m3p.y.x == 1:3
+end
+
+@testset "functor(typeof(x), y) from @functor" begin
+  nt1, re1 = functor(Foo, (x=1, y=2, z=3))
+  @test nt1 == (x = 1, y = 2)
+  @test re1((x = 10, y = 20)) == Foo(10, 20)
+  re1((y = 22, x = 11)) # gives Foo(22, 11), is that a bug?
+
+  nt2, re2 = functor(Foo, (z=33, x=1, y=2))
+  @test nt2 == (x = 1, y = 2)
+  @test re2((x = 10, y = 20)) == Foo(10, 20)
+
+  @test_throws Exception functor(Foo, (z=33, x=1))  # type NamedTuple has no field y
+
+  nt3, re3 = functor(OneChild3, (x=1, y=2, z=3))
+  @test nt3 == (y = 2,)
+  @test re3((y = 20,)) == OneChild3(1, 20, 3)
+  re3(22) # gives OneChild3(1, 22, 3), is that a bug?
+end
+
+@testset "functor(typeof(x), y) for Base types" begin
+  nt11, re11 = functor(NamedTuple{(:x, :y)}, (x=1, y=2, z=3))
+  @test nt11 == (x = 1, y = 2)
+  @test re11((x = 10, y = 20)) == (x = 10, y = 20)
+  re11((y = 22, x = 11))
+  re11((11, 22))  # passes right through
+
+  nt12, re12 = functor(NamedTuple{(:x, :y)}, (z=33, x=1, y=2))
+  @test nt12 == (x = 1, y = 2)
+  @test re12((x = 10, y = 20)) == (x = 10, y = 20)
+
+  @test_throws Exception functor(NamedTuple{(:x, :y)}, (z=33, x=1))
+end
+
+###
+### Extras
+###
+
 @testset "Walk" begin
   model = Foo((0, Bar([1, 2, 3])), [4, 5])
 
@@ -60,13 +122,6 @@ end
   @test model′ == (; x=(0, (; x=[1, 2, 3])), y=[4, 5])
 end
 
-@testset "Property list" begin
-  model = Baz(1, 2, 3)
-  model′ = fmap(x -> 2x, model)
-  
-  @test (model′.x, model′.y, model′.z) == (1, 4, 3)
-end
-
 @testset "fcollect" begin
   m1 = [1, 2, 3]
   m2 = 1
@@ -78,7 +133,7 @@ end
 
   m1 = [1, 2, 3]
   m2 = Bar(m1)
-  m0 = NoChildren(:a, :b)
+  m0 = NoChildren2(:a, :b)
   m3 = Foo(m2, m0)
   m4 = Bar(m3)
   @test all(fcollect(m4) .=== [m4, m3, m2, m1, m0])
@@ -89,6 +144,79 @@ end
   @test all(fcollect(m3) .=== [m3, m1, m2])
 end
 
+###
+### Vararg forms
+###
+
+@testset "fmap(f, x, y)" begin
+  m1 = (x = [1,2], y = 3)
+  n1 = (x = [4,5], y = 6)
+  @test fmap(+, m1, n1) == (x = [5, 7], y = 9)
+
+  # Reconstruction type comes from the first argument
+  foo1 = Foo([7,8], 9)
+  @test fmap(+, m1, foo1) == (x = [8, 10], y = 12)
+  @test fmap(+, foo1, n1) isa Foo
+  @test fmap(+, foo1, n1).x == [11, 13]
+
+  # Mismatched trees should be an error
+  m2 = (x = [1,2], y = (a = [3,4], b = 5))
+  n2 = (x = [6,7], y = 8)
+  @test_throws Exception fmap(first∘tuple, m2, n2)  # ERROR: type Int64 has no field a
+  @test_throws Exception fmap(first∘tuple, m2, n2)
+
+  # The cache uses IDs from the first argument
+  shared = [1,2,3]
+  m3 = (x = shared, y = [4,5,6], z = shared)
+  n3 = (x = shared, y = shared, z = [7,8,9])
+  @test fmap(+, m3, n3) == (x = [2, 4, 6], y = [5, 7, 9], z = [2, 4, 6])
+  z3 = fmap(+, m3, n3)
+  @test z3.x === z3.z
+
+  # Pruning of duplicates:
+  @test fmap(+, m3, n3; prune = nothing) == (x = [2,4,6], y = [5,7,9], z = nothing)
+
+  # More than two arguments:
+  z4 = fmap(+, m3, n3, m3, n3)
+  @test z4 == fmap(x -> 2x, z3)
+  @test z4.x === z4.z
+
+  @test fmap(+, foo1, m1, n1) isa Foo
+  @static if VERSION >= v"1.6" # fails on Julia 1.0
+    @test fmap(.*, m1, foo1, n1) == (x = [4*7, 2*5*8], y = 3*6*9)
+  end
+end
+
+@static if VERSION >= v"1.6" # Julia 1.0: LoadError: error compiling top-level scope: type definition not allowed inside a local scope
+@testset "old test update.jl" begin
+  struct M{F,T,S}
+    σ::F
+    W::T
+    b::S
+  end
+
+  @functor M
+
+  (m::M)(x) = m.σ.(m.W * x .+ m.b)
+
+  m = M(identity, ones(Float32, 3, 4), zeros(Float32, 3))
+  x = ones(Float32, 4, 2)
+  m̄, _ = gradient((m,x) -> sum(m(x)), m, x)
+  m̂ = Functors.fmap(m, m̄) do x, y
+    isnothing(x) && return y
+    isnothing(y) && return x
+    x .- 0.1f0 .* y
+  end
+
+  @test m̂.W ≈ fill(0.8f0, size(m.W))
+  @test m̂.b ≈ fill(-0.2f0, size(m.b))
+end
+end  # VERSION
+
+###
+### FlexibleFunctors.jl
+###
+
 struct FFoo
   x
   y
@@ -102,13 +230,13 @@ struct FBar
 end
 @flexiblefunctor FBar p
 
-struct FBaz
+struct FOneChild4
   x
   y
   z
   p
 end
-@flexiblefunctor FBaz p
+@flexiblefunctor FOneChild4 p
 
 @testset "Flexible Nested" begin
   model = FBar(FFoo(1, [1, 2, 3], (:y, )), (:x,))
@@ -132,7 +260,7 @@ end
 end
 
 @testset "Flexible Property list" begin
-  model = FBaz(1, 2, 3, (:x, :z))
+  model = FOneChild4(1, 2, 3, (:x, :z))
   model′ = fmap(x -> 2x, model)
 
   @test (model′.x, model′.y, model′.z) == (2, 2, 6)
@@ -147,7 +275,7 @@ end
   @test all(fcollect(m4, exclude = x -> x isa Array) .=== [m4, m3])
   @test all(fcollect(m4, exclude = x -> x isa FFoo) .=== [m4])
 
-  m0 = NoChildren(:a, :b)
+  m0 = NoChildren2(:a, :b)
   m1 = [1, 2, 3]
   m2 = FBar(m1, ())
   m3 = FFoo(m2, m0, (:x, :y,))
diff --git a/test/runtests.jl b/test/runtests.jl
index 394cfbf..5b853fc 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -5,7 +5,6 @@ using Zygote
 
   include("basics.jl")
   include("base.jl")
-  include("update.jl")
 
   if VERSION < v"1.6" # || VERSION > v"1.7-"
     @warn "skipping doctests, on Julia $VERSION"
diff --git a/test/update.jl b/test/update.jl
deleted file mode 100644
index 0ed6bca..0000000
--- a/test/update.jl
+++ /dev/null
@@ -1,23 +0,0 @@
-@testset "Generalized fmap over equivalent functors" begin
-  struct M{F,T,S}
-    σ::F
-    W::T
-    b::S
-  end
-
-  @functor M
-
-  (m::M)(x) = m.σ.(m.W * x .+ m.b)
-
-  m = M(identity, ones(Float32, 3, 4), zeros(Float32, 3))
-  x = ones(Float32, 4, 2)
-  m̄, _ = gradient((m,x) -> sum(m(x)), m, x)
-  m̂ = Functors.fmap(m, m̄) do x, y
-    isnothing(x) && return y
-    isnothing(y) && return x
-    x .- 0.1f0 .* y
-  end
-
-  @test m̂.W ≈ fill(0.8f0, size(m.W))
-  @test m̂.b ≈ fill(-0.2f0, size(m.b))
-end