In this talk, I will discuss the front propagation for solutions of one-dimensional diffusion equations on the whole line: u_t = u_{xx} + f(x,u). Here f(x,u) is a smooth function that is periodic in x and satisfies f(x,0) = 0. We consider a large class of nonlinearities f including multistable ones. Our analysis reveals some new dynamics where the asymptotic profile of the solution is not characterized by a single front, but by a layer of several fronts traveling at different speeds, which we call a ``propagating terrace". This is joint work with Thomas Giletti and Arnaud Ducrot.