diff --git a/Project.toml b/Project.toml
index 30a618092..b8abe79e4 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,6 +1,6 @@
 name = "ChainRulesCore"
 uuid = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
-version = "1.26.7"
+version = "1.27.0"
 
 [deps]
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
diff --git a/src/tangent_types/structural_tangent.jl b/src/tangent_types/structural_tangent.jl
index 94a0e4e48..0e5a4d3dc 100644
--- a/src/tangent_types/structural_tangent.jl
+++ b/src/tangent_types/structural_tangent.jl
@@ -405,6 +405,14 @@ function Base.indexed_iterate(tangent::Tangent{P,<:Tuple}, i::Int, state=1) wher
     return Base.indexed_iterate(backing(tangent), i, state)
 end
 
+# Whether a structural tangent should broadcast as a container (e.g. a tuple broadcast) or
+# as a scalar (e.g. a 0-dim broadcast with a struct-valued output) depends on the broadcast,
+# and in general can't be determined from the tangent alone (e.g. for `Tangent{Any}`)
+function Base.Broadcast.broadcastable(::StructuralTangent)
+    msg = "broadcasting over structural tangents is not supported; use `map` or arithmetic"
+    return throw(ArgumentError(msg))
+end
+
 function Base.map(f, tangent::Tangent{P,<:Tuple}) where {P}
     vals::Tuple = map(f, backing(tangent))
     return Tangent{P,typeof(vals)}(vals)
diff --git a/test/tangent_types/structural_tangent.jl b/test/tangent_types/structural_tangent.jl
index c0b26217e..746c55849 100644
--- a/test/tangent_types/structural_tangent.jl
+++ b/test/tangent_types/structural_tangent.jl
@@ -140,6 +140,15 @@ end
             @inferred _unpack2tuple(ctup)
             @test _unpack2tuple(ctup) === (2.0, 3)
 
+            # Test broadcasting is not supported
+            @test_throws ArgumentError ctup .* 2
+            @test_throws ArgumentError Tangent{Any}(1.0, [2.0]) .* 2
+            @test_throws ArgumentError ctup .+ [1, 2]
+            @test_throws ArgumentError Tangent{Foo}(; y=2.0, x=1.0) .* 2
+            @test_throws ArgumentError Tangent{Dict}(Dict(:a => 1.0)) .* 2
+            @test_throws ArgumentError MutableTangent{MFoo}(; x=1.5, y=1.5) .* 2
+            @test_throws ArgumentError @thunk(ctup) .* 2
+
             # Test getproperty is inferrable
             _unpacknamedtuple = tangent -> (tangent.x, tangent.y)
             @inferred _unpacknamedtuple(Tangent{Foo}(; x=2, y=3.0))