  These are notes of lectures given at the NATO Summer School, Montreal 1995.
Taubes's recent spectacular work setting up a correspondence between
$J$-holomorphic curves in symplectic 4-manifolds and solutions of the
Seiberg-Witten equations counts $J$-holomorphic curves in a somewhat new way.
The "standard" theory concerns itself with moduli spaces of connected curves,
and gives rise to Gromov-Witten invariants. However, Taubes's curves arise as
zero sets of sections and so need not be connected. These notes are in the main
expository. We first discuss the invariants as Taubes defined them, and then
discuss some alternatives, showing, for example, a way of dealing with
multiply-covered exceptional spheres. We also calculate some examples, in
particular finding the Gromov invariant of the fiber class of an elliptic
surface by counting $J$-holomorphic curves, rather than going via
Seiberg--Witten theory.
