  In this paper we present a new approach to Morse theory based on the de
Rham-Federer theory of currents. The full classical theory is derived in a
transparent way. The methods carry over uniformly to the equivariant and the
holomorphic settings. Moreover, the methods are substantially stronger than the
classical ones and have interesting applications to geometry. They lead, for
example, to formulae relating characteristic forms and singularities of bundle
maps.
