Piecewise-constant layered media have long served as a basic model for seismic imaging. From a mathematical point of view they are of interest as a case study in PDE with discontinuous coefficients. Although the plane wave impulse response at the boundary for layered media has been studied extensively since the 1950s, an exact formula was discovered only recently. In this talk we describe how this formula provides new insight into both theoretical and practical aspects of the standard inverse problem, and reveals some remarkable mathematical structure not previously known to relate to layered media.