  In this lecture I review recent results on the first order equations
describing BPS extremal states, in particular N=2 extremal black-holes. The
role of special geometry is emphasized also in the rigid theory and a
comparison is drawn with the supersymmetric derivation of instantons and
hyperinstantons in topological field theories. Work in progress on the
application of solvable Lie algebras to the discussion of BPS states in
maximally extended supergravities is outlined.
