We compute the open Gromov-Witten invariants for every compact semi-Fano toric surface, i.e. a toric surface X with nef anticanonical bundle. Unlike the Fano case, this involves non-trivial obstructions in the corresponding moduli problem.
As an application, an explicit expression of the superpotential W for the mirror of X is obtained, which in turn gives an explicit ring presentation of the small quantum cohomology of X. We also give a computational verification of the natural ring isomorphism between the small quantum cohomology of X and the Jacobian ring of W.