The mixture model has long been a challenge to the statistician, whether beginner, practitioner or theoretician. Recent times have seen great advances in our understanding of the some basic mathematical features of this model, and these notes are meant to be a unification of the work I have carried out, jointly with many wonderful collaborators, in this area. Based on lectures given in 1993 at a regional conference of the Conference Board of the Mathematical Sciences, the notes are directed toward a mixed audience of advanced graduate students and research workers in this and related areas. For the sake of newcomers to the mixture model, I will attempt to be complete enough for the text to make sense in itself, but must at some points refer the reader to other more extensive treatments.