I will be concerned with initial-boundary value problems for systems of conservation laws in one space variable. First, I will go over some of the most relevant features of these problems. In particular, I will stress that different viscous approximation lead, in general, to different limits.
Next, I will discuss possible ways of characterizing the limit of a given viscous approximation. Also, I will establish a uniqueness criterion that allows to conclude that the limit of a self-similar approximation introduced by Dafermos et al. actually coincide with the limit of the physical viscous approximation. Finally, if time allows I will mention consequences on the design of numerical schemes. The talk will be based on joint works with S. Bianchini, C. Christoforou and S. Mishra.