We begin by a general overview of some long standing results on the characteriza- tion of global attractors for dynamical systems generated by scalar semilinear parabolic equations defined on a interval under separated boundary conditions. We then survey some recent results that extend the previous characterization to the case of periodic boundary conditions. We point out that under separated boundary conditions the gener- ated flow has a variational character and the global attractor has a Morse decomposition, while the case of periodic boundary conditions is, in general, non variational and the flow may possess periodic orbits.