A caustic is the singular value set of a Lagrangian map — the pattern a flow of shortest paths makes where it focuses. The same short universal list of local shapes (Arnol’d’s fold, cusp, swallowtail, umbilic) turns up in optics, in the sub-Riemannian conjugate locus of a Lie group, and in the shell-crossing walls of the cosmic web. These articles treat caustics both as objects to compute forward and as data to invert — reading the hidden geometry back out of the focusing pattern.