diff --git a/src/Optimisers.jl b/src/Optimisers.jl
index 417b90d4..616f0ea2 100644
--- a/src/Optimisers.jl
+++ b/src/Optimisers.jl
@@ -2,6 +2,7 @@ module Optimisers
 
 using Functors: functor, fmap, isleaf
 using LinearAlgebra
+using Base.Broadcast: broadcast_preserving_zero_d, broadcasted
 
 include("interface.jl")
 
diff --git a/src/interface.jl b/src/interface.jl
index 235c2e94..d4b0b542 100644
--- a/src/interface.jl
+++ b/src/interface.jl
@@ -21,7 +21,13 @@ function setup(rule, x; seen = Base.IdSet())
   end
 end
 
-subtract!(x, x̄) = iswriteable(x) ? (x .= x .- x̄) : eltype(x).(x .- x̄)
+function subtract!(x, x̄)
+  if iswriteable(x)
+    x .= x .- x̄
+  else
+    broadcast_preserving_zero_d(eltype(x), broadcasted(-, x, x̄))
+  end
+end
 
 update!(::Nothing, x, ::Zero, ::Zero...) = nothing, x
 update!(::Nothing, x, x̄s...) = nothing, x
diff --git a/test/rules.jl b/test/rules.jl
index ffb4ca65..63aef5bb 100644
--- a/test/rules.jl
+++ b/test/rules.jl
@@ -226,3 +226,31 @@ end
     @test static_loss(static_model) < 1.9 
   end
 end
+
+@testset "zero-dim arrays" begin
+  empty!(LOG)
+  @testset "$(name(o))" for o in RULES
+    m = (; arr = fill(1.0), pda = PermutedDimsArray(fill(1.0), ()), ref = Ref(1.0))
+    # The point of PermutedDimsArray here is to test the out-of-place path, so check:
+    @test Optimisers.iswriteable(m.arr)
+    @test !Optimisers.iswriteable(m.pda)
+    s = Optimisers.setup(o, m)
+    for _ in 1:10^3
+      g = loggradient(o)(x -> abs2(first(x.arr) + first(x.pda) + first(x.ref)), m)[1]
+      s, m = Optimisers.update(s, m, g)
+    end
+    # Goal is to check that broadcasting does not accidentally make a scalar,
+    # but `m.arr` iscopied &  mutated, so only `m.pda` is a real test:
+    @test m.arr isa Array{Float64, 0}
+    @test m.pda isa AbstractArray{Float64, 0}
+    @test m.ref isa Ref  # because it's mutated, broadcast_preserving_zero_d would make an array
+    if o isa RADAM
+      @test sum(m.arr) < 0.7
+      @test_broken sum(m.arr) < 0.3
+    else
+      @test sum(m.arr) < 0.3
+      @test sum(m.pda) < 0.3
+    end
+    @test only(m.ref) ≈ 1  # not currently regarded as trainable
+  end
+end
diff --git a/test/runtests.jl b/test/runtests.jl
index d47bce08..e01ca411 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -80,7 +80,7 @@ Optimisers.trainable(x::TwoThirds) = (a = x.a,)
     end
 
     @testset "trainable subset" begin
-      @info "ignore these warnings about trainable, testing the old path"
+      @info "ignore these warnings about `trainable`, they are testing the path for old-style methods"
       # Foo has an old-style tuple trainable, both elements
       mf = Foo([1.0, 2.0], (a = sin, b = [3.0, 4.0], c = 5))
       sf = Optimisers.setup(Descent(0.1), mf)
@@ -131,6 +131,30 @@ Optimisers.trainable(x::TwoThirds) = (a = x.a,)
       @test eltype(m4[2]) == Float32
     end
 
+    @testset "zero dimension" begin
+      # Mutable Array{T,0}
+      m = fill(1.0)
+      s = Optimisers.setup(Descent(0.1), m)
+      s2, m2 = Optimisers.update!(s, m, fill(2.0))
+      @test m2 === m
+      @test only(m) ≈ 0.8
+
+      # "Immutable" zero-array, takes out-of-place path:
+      m3 = PermutedDimsArray(fill(1.0), ())
+      @test !Optimisers.iswriteable(m3)  # note that there's Base.iswritable, it seems I can't spell
+      s3 = Optimisers.setup(Descent(0.1), m3)
+      s4, m4 = Optimisers.update!(s3, m3, fill(2.0))
+      @test m4 !== m3
+      @test only(m4) ≈ 0.8
+
+      # Ref, should this be regarded as holding a parameter? At present it's not:
+      m5 = Ref(1.0)
+      s5 = Optimisers.setup(Descent(0.1), m5)
+      g5 = gradient(m -> m[]^2, m5)[1]  # (x = 2.0,)
+      s6, m6 = Optimisers.update!(s5, m5, g5)
+      @test m6[] ≈ 1
+    end
+
     @testset "forgotten gradient" begin
       x = [1.0, 2.0]
       sx = Optimisers.setup(Descent(), x)