I will discuss some recent work on shock formation and propagation of singularities for compressible Euler.
The aim is to describe precisely the structure of an entropy producing shock in its early phase (starting from
smooth initial conditions), and also describe with some detail a collection of weaker singularities that are born
with it. These singularities are in the derivatives of the solution fields and travel along different characteristics
than the shock.  This reports on joint work with T. Buckmaster, S. Shkoller, and V. Vicol.