Abstract: Spectral theory is the study of the relation between the data defining a problem (typically the coefficients in a differential or difference equation) and the spectrum associated to that problem. By gems I mean necessary and sufficient conditions that set up a bijection between a class of equation data and a class of spectral data. I'll focus on a collection of results in the simplest of spectral problems - those defined by orthogonal polynomials. I'll discuss a class of nonlinear analogs of the Plancherel theorem that go back to 1920 with results as recent as 2006.