In the last two decades, there have been dramatic advances in the rigorous mathematical theory of shock waves in solutions to the relativistic Euler equations and their non-relativistic analog, the compressible Euler equations. A lot of the progress has relied on techniques that were developed to study Einstein’s equations. In this talk, I will provide an overview of the field and highlight some recent progress, with a focus on results that reveal various aspects of the structure of the maximal development of the data and the corresponding implications for the shock development problem, which is the problem of continuing the solution weakly after a shock. I will also describe various open problems, some of which are tied to the Einstein–Euler equations. Various aspects of this program are joint with L. Abbrescia, M. Disconzi, and J. Luk.