Topological version of the Wiener-Wintner theorem characterizes $\lambda$ on complex unit circle for which averages $\frac{1}{N} \sum \limits_{n=0}^{N-1} \lambda^n f(T^n x)$ are uniformly convergent for every continuous function $f \in C(X)$, where $(X,T)$ is a topological dynamical system. I will present extensions of this result to amenable semigroups of Markov operators on $C(X)$. This is joint work with W. Bartoszek.