Our talk has its starting point in the classical problem of enumerating curves
constrained by generic incidence conditions in a projective variety.
Over the complex numbers we have at our disposal a powerful theory
centered around Gromov-Witten invariants. Over the reals, we witnessed
a breakthrough in 2003 when Welschinger discovered new invariants
adapted to the count of real rational curves inside $2$\yh- and $3$\yh-dimensional
varieties. We will explain the construction of these invariants and will give
a survey of related results.