The subject of enumerative geometry goes back at least to the middle
of the 19th centuary. It deals with questions of enumerating geometric
objects, e.g.
(a) how many lines pass through 2 points or
through 1 point and 2 lines in 3-space?
(b) how many conics in 2-space are tangent to k lines and
pass through 5-k points?

There has been an explosition of activity in this field over the past
twenty years, following the development of Gromov-Witten invariants in
sympletic topology and string theory. The idea of counting parameterizations
of curves in order to count curves themselves has led to solutions of
whole sets of long-standing classical problems. At the same time, string
theory has generated a multitude of predictions for the structure of
GW-invariants, as well as for the behavior of certain natural families
of Laplacians. It has in particular suggested that there is a diality
between certain symplectic and complex manifolds and that in some cases
GW-invariants see some geometric objects, that are yet to be fully
discovered mathematically.

In this talk I hope to give an indication of what enumerative geometry
is about and of the shift in the paradigm that has occured over the past
two decades.