After recalling some existence and uniqueness results for systems of
conservation laws in one space variable, I will focus on the regularity
properties of the admissible solution.  It is known that, even in the case
of a single equation, classical solutions  can break down in finite time
owing to the formation of discontinuities.  As a consequence, one can only
hope for very mild regularity results.

In particular, in 1973 Schaeffer established a result that applies to
scalar conservation laws with
convex fluxes and can be loosely speaking formulated as follows: for a
generic smooth initial datum,
 the admissible solution is smooth outside a locally finite number of
curves in the (t, x) plane. Here "generic" should be interpreted in a
suitable sense,  related to the Baire Category Theorem.  My talk will aim
at discussing a recent counter-example that rules out the possibility of
extending Schaeffer's Theorem to systems of conservation laws.
The talk will be based on a joint work with Laura Caravenna.