The symmetry of solutions of elliptic equations is a classical and challenging problem in PDE, connected with stability. In this talk we are concerned with parabolic equations and we ask whether the 1-dimensional symmetry eventually emerges in the long time, for solutions which are initially non-symmetric. We will present a satisfactory answer in the case of the Fisher-KPP equation, together with some counter-examples and open questions. This topic is the object of a joint work with F. Hamel.