We review how transport properties for chaotic dynamical systems may be studied through cycle expan- expansions, and show how anomalies can be quantitatively described by hierarchical sequences of periodic orbits. In this paper we consider deterministic dynamical systems that exhibit nontrivial transport properties (which may be associated to either normal or anomalous diffusion). We start by dealing with the case of hyperbolic systems where typically normal diffusion is observed (even though actual calculation of transport coefficients may be exceedingly difficult [1,2]), while the in the second part we consider weakly chaotic systems, where long trappings near regular phase-space regions may induce anomalies in diffusive properties.