Overdetermined systems, conformal geometry, and the BGG complex

This is an expanded version of a series of two lectures given at the IMA summer program "Symmetries and Overdetermined Systems of Partial Differential Equations". The main part of the article describes the Riemannian version of the prolongation procedure for certain overdetermined system obtained recently in joint work with T.P. Branson, M.G. Eastwood, and A.R. Gover. First a simple special case is discussed, then the (Riemannian) procedure is described in general. The prolongation procedure was derived from a simplification of the construction of Bernstein-Gelfand-Gelfand (BGG) sequences of invariant differential operators for certain geometric structures. The version of this construction for conformal structures is described next. Finally, we discuss generalizations of both the prolongation procedure and the construction of invariant operators to other geometric structures.