I will describe my recent works, some joint with M. Disconzi and J. Luk, on the compressible Euler equations and their relativistic analog. The starting point is new formulations of the equations exhibiting miraculous geo-analytic structures, including i) a sharp decomposition of the flow into geometric wave and transport-div-curl parts, ii) null form source terms, and iii) structures that allow one to propagate one additional degree of differentiability (compared to standard estimates) for the entropy and vorticity. I will then describe a main application: the study of stable shock formation, without symmetry assumptions, in more than one spatial dimension. I will emphasize the role that nonlinear geometric optics plays in the analysis and highlight how the new formulations allow for its implementation. Finally, I will describe some important open problems, and I will connect the results to the broader goal of obtaining a rigorous mathematical theory that models the long-time behavior of solutions in the presence of shock singularities.