This thesis is concerned with the properties of a number of selected processes taking place on complex networks and the way they are affected by structure and evolution of the networks. What is meant here by 'complex networks' is the graph-theoretical representations and models of various empirical networks (e.g., the Internet network) which contain both random and deterministic structures, and are characterised among others by the small-world phenomenon, power-law vertex degree distributions, or modular and hierarchical structure. The mathematical models of the processes taking place on these networks include percolation and random walks we utilise.The results presented in the thesis are based on five thematically coherent papers. The subject of the first paper is calculating thresholds for epidemic outbreaks on dynamic networks, where the disease spread is modelled by percolation. In the paper, known analytical solutions for the epidemic thresholds were extended to a class of dynamically evolving networks; additionally, the effects of finite size of the network on the magnitude of the epidemic were studied numerically. The subject of the second and third paper is the static and dynamic properties of two diametrically opposed random walks on model highly symmetric deterministic graphs. Specifically, we analytically and numerically find the stationary states and relaxation times of the ordinary, diffusive random walk and the maximal-entropy random walk. The results provide insight into localisation of random walks or their trapping in isolated regions of networks. Finally, in the fourth and fifth paper, we examine the utility of random walks in detecting topological features of complex networks. In particular, we study properties of the centrality measures (roughly speaking, the ranking of vertices) based on random walks, as well as we conduct a systematic comparative study of random-walk based methods of detecting modular structure of networks.These studies thus aimed at specific problems in modelling and analysis of complex networks, including theoretical examination of the ways the behaviour of random processes intertwines with the structure of complex networks.