  We devote this paper to the issue of existence of pulsating travelling front
solutions for spatially periodic heterogeneous reaction-diffusion equations in
arbitrary dimension, in both bistable and more general multistable frameworks.
In the multistable case, the notion of a single front is not sufficient to
understand the dynamics of solutions, 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 whose speeds are
ordered. Surprisingly, for a given equation, the shape of this terrace (i.e.,
the involved intermediate states or even the cardinality of the family of
fronts) may depend on the direction of propagation.
