I will present some of our results on the general topic of random walks on networks. In particular, I will discuss the results about how heterogeneities influence the dynamics of, so-called, continuous time random walk model. Then I will talk about some further applications of the heterogeneous random walk model to the epidemiological models on static and temporal networks. Moreover, I will present some results of the discrete-state models on small graphs and discuss the challenges of the presented methods applications to the real data.