Enumerative geometry of curves: old and new

I will present an overview of the (complex) Gromov-Witten invariants and their relation to curve counts provided by Pandharipande's version Gopakumar-Vafa formula for Fano classes. I will then present a similar formula that transforms the real positive-genus GW-invariants of many real-orientable threefolds into signed counts of curves. These integer invariants provide lower bounds for counts of real curves of a given genus that pass through conjugate pairs of constraints. This talk is based on recent joint works with P. Georgieva and J. Niu.