this talk will be devoted to the existence of pulsating travelling front solutions for spatially periodic heterogeneous reaction-diffusion equations in arbitrary dimension. in the bistable case, such a pulsating front indeed exists and it also describes the large time dynamics of solutions of the cauchy problem. however, unlike in the homogeneous case the periodic problem is no longer invariant by rotation, so that the front speed may be different depending on its direction. this in turn raises some difficulties in the spreading shape of solutions of the evolution problem, which may exhibit strongly asymmetrical features. in the general multistable case, that is when there is a finite but arbitrary number of stable steady states, the notion of a single front is no longer sufficient and we instead observe the appearance of a so-called propagating terrace. this roughly refers to a finite family of stacked fronts connecting intermediate stable steady states and whose speeds are ordered. the presented results come from a series of work with w. ding, a. ducrot, h. matano and l. rossi.