  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.
