diff --git a/src/adjtrans.jl b/src/adjtrans.jl
index 2c144903..b8b0efd7 100644
--- a/src/adjtrans.jl
+++ b/src/adjtrans.jl
@@ -1,4 +1,5 @@
-export AdjointLinearOperator, TransposeLinearOperator, adjoint, transpose
+export AdjointLinearOperator, TransposeLinearOperator, ConjugateLinearOperator,
+       adjoint, transpose, conj
 
 # From julialang:stdlib/LinearAlgebra/src/adjtrans.jl
 struct AdjointLinearOperator{T} <: AbstractLinearOperator{T}
@@ -37,6 +38,7 @@ size(A :: ConjugateLinearOperator, d :: Int) = size(A.parent, d)
 for f in [:hermitian, :ishermitian, :symmetric, :issymmetric]
   @eval begin
     $f(A :: AdjTrans) = $f(A.parent)
+    $f(A :: ConjugateLinearOperator) = $f(A.parent)
   end
 end
 
@@ -89,14 +91,17 @@ end
 
 function *(op :: ConjugateLinearOperator, v :: AbstractVector)
   p = op.parent
-  return conj.(p * v)
+  return conj.(p * conj.(v))
 end
 
--(op :: AdjointLinearOperator) = adjoint(-op.parent)
+-(op :: AdjointLinearOperator)   = adjoint(-op.parent)
 -(op :: TransposeLinearOperator) = transpose(-op.parent)
+-(op :: ConjugateLinearOperator) = conj(-op.parent)
 
-*(op :: AdjointLinearOperator, x :: Number) = adjoint(op.parent * x)
+*(op :: AdjointLinearOperator, x :: Number)   = adjoint(op.parent * conj(x))
 *(op :: TransposeLinearOperator, x :: Number) = transpose(op.parent * x)
+*(op :: ConjugateLinearOperator, x :: Number) = conj(op.parent * conj(x))
 
-*(x :: Number, op :: AdjointLinearOperator) = adjoint(x * op.parent)
+*(x :: Number, op :: AdjointLinearOperator)   = adjoint(conj(x) * op.parent)
 *(x :: Number, op :: TransposeLinearOperator) = transpose(x * op.parent)
+*(x :: Number, op :: ConjugateLinearOperator) = conj(conj(x) * op.parent)
diff --git a/test/runtests.jl b/test/runtests.jl
index e0bfcb76..24e7aaa8 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -3,6 +3,7 @@ using Test, TestSetExtensions, LinearOperators
 include("test_aux.jl")
 
 include("test_linop.jl")
+include("test_adjtrans.jl")
 include("test_cat.jl")
 include("test_lbfgs.jl")
 include("test_lsr1.jl")
diff --git a/test/test_adjtrans.jl b/test/test_adjtrans.jl
new file mode 100644
index 00000000..a8f5440c
--- /dev/null
+++ b/test/test_adjtrans.jl
@@ -0,0 +1,38 @@
+function test_adjtrans()
+  @testset ExtendedTestSet "Adjoint/Transpose/Conjugate" begin
+    A = rand(5,3) + im * rand(5,3)
+    opA = LinearOperator(A)
+
+    aopA = AdjointLinearOperator(opA)
+    copA = ConjugateLinearOperator(opA)
+    topA = TransposeLinearOperator(opA)
+
+    for (foo,fop) in [(adjoint, aopA),
+                      (conj, copA),
+                      (transpose, topA)]
+      @test foo(opA) === fop
+      @test Matrix(fop) == foo(A)
+      @test foo(fop) === opA
+
+      @test Matrix(-fop)    == foo(-A)
+      @test Matrix((2 + 3im) * fop) == (2 + 3im) * foo(A)
+      @test Matrix(fop * (2 + 3im)) == foo(A) * (2 + 3im)
+    end
+
+    @test adjoint(topA)   === copA
+    @test adjoint(copA)   === topA
+    @test conj(aopA)      === topA
+    @test conj(topA)      === aopA
+    @test transpose(copA) === aopA
+    @test transpose(aopA) === copA
+
+    v = rand(5) + im * rand(5)
+    @test aopA * v == adjoint(A) * v
+    @test topA * v == transpose(A) * v
+
+    v = rand(3) + im * rand(3)
+    @test copA * v == conj(A) * v
+  end
+end
+
+test_adjtrans()
diff --git a/test/test_linop.jl b/test/test_linop.jl
index 8907329c..0915a7e9 100644
--- a/test/test_linop.jl
+++ b/test/test_linop.jl
@@ -7,6 +7,7 @@ function test_linop()
   @testset ExtendedTestSet "Basic operations" begin
     for op = (LinearOperator(A1), LinearOperator(A1')',
               transpose(LinearOperator(transpose(A1))),
+              conj(LinearOperator(conj(A1))),
               PreallocatedLinearOperator(A1))
       show(op);
 
@@ -437,6 +438,13 @@ function test_linop()
     A = simple_matrix(ComplexF64, 5, 5)
     @test_throws LinearOperatorException opCholesky(A, check=true)  # Not Hermitian / positive definite
     @test_throws LinearOperatorException opCholesky(-A'*A, check=true)  # Not positive definite
+
+    # Adjoint of a symmetric non-hermitian
+    A = simple_matrix(ComplexF64, 3, 3)
+    A = A + transpose(A)
+    op = LinearOperator(3, 3, true, false, v -> A * v)
+    v = rand(3)
+    @test op' * v ≈ A' * v
   end
 
   @testset ExtendedTestSet "Type specific operator" begin