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Tensor operators are discussed for Hopf algebras and, in particular, for a quantum (q-deformed) algebra Uq(g), where g is any simple finite-dimensional or affine Lie algebra. These operators are defined via an adjoint action in a Hopf algebra. There are two types of the tensor operators which correspond to two coproducts in the Hopf algebra. In the case of tensor products of two tensor operators one can obtain 8 types of the tensor operators and so on. We prove the relations which can be a basis for a proof of the Wigner-Eckart theorem for the Hopf algebras. It is also shown that in the case of Uq(g) a scalar operator can be differed from an invariant operator but at q=1 these operators coincide.