We study the long-time behavior of scalar viscous conservation laws via the structure of ω-limit sets. We show that ω-limit sets always contain constants or shocks by establishing convergence to shocks for arbitrary monotone initial data. In the particular case of Burgers' equation, we review and refine results that parametrize entire solutions in terms of probability measures, and we construct initial data for which the ω-limit set is not reduced to the translates of a single shock. Finally we propose several open problems related to the description of long-time dynamics.