We present recent results on the characterization of the global semiflow generated by scalar semilinear parabolic equations of the form ut​=uxx​+f(u,ux​), x∈S1, t≥0, with periodic boundary conditions. Due to the S1-equivariance property provided by the nonlinearity f=f(u,ux​), the global attractor consists only of constant equilibria, standing waves, rotating waves and heteroclinic orbits connecting them. Moreover, when all the critical elements are hyperbolic, the global attractor has the Morse-Smale property. This fact provides a braid type characterization of the global attractor allowing the determination of all the heteroclinic connections occurring in the attractor.