  We study the long-time behavior of scalar viscous conservation laws via the
structure of $\omega$-limit sets. We show that $\omega$-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 $\omega$-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.
