This paper summarizes the general theory and properties for Apodized Pupil Lyot Coronagraphs which consist of a classical hard-edged Lyot coronagraph with an upstream pupil apodization. The ideal apodization function can be determined from an integral eigenvalue problem which solutions are prolate spheroidal functions. Solutions exist for any geometrty, including rectangular, circular, or elliptical. Formal solutions can be extended to the case of arbitrary apertures, using generalized prolate spheroidal functions for centrally obstructed apertures, spiders, or segmented telescopes. The properties of these coronagraphs enable the possibility of multiple stage coronagraphy and achromatization. The new instrument Gemini Planet Imager (GPI) will include such a coronagraph.