The talk presents a new formula for the Gromov-Witten invariants ofarbitrary genus in the projective plane as well as for the relatedenumerative invariants in other toric surfaces. The answer is given interms of certain lattice paths in the relevant Newton polygon. Thelength of the paths turns out to be responsible for the genus of theholomorphic curves in the count. The formula is obtained by working interms of the so-called tropical algebraic geometry. This version ofalgebraic geometry is simpler than its classical counterpart in manyaspects. In particular, complex algebraic varieties themselves becomepiecewise-linear objects in the real space. The transition from theclassical geometry is provided by consideration of the "large complexlimit" (which is also known as "dequantization" or "patchworking" insome other areas of Mathematics).