  We compute Gromov-Witten invariants of any genus for del Pezzo surfaces of
degree $\ge2$. The genus zero invariants have been computed a long ago,
Gromov-Witten invariants of any genus for del Pezzo surfaces of degree $\ge3$
have been found by Vakil. We solve the problem in two steps: (1) we consider
the plane blown up at 6 points on a conic and one more point outside this conic
(weak Fano surface), and, using techniques of tropical geometry, obtain a
Caporaso-Harris type formula counting curves of any divisor class and genus
subject to arbitrary tangency conditions with respect to the blown up conic,
(2) then we express the Gromov-Witten invariants of the plane blown up at 7
points via enumerative invariants of the weak Fano surface, using Vakil's
version of Abramovich-Bertram formula.

Algebraic Geometry