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In this paper, we compute the open Gromov–Witten invariants for every compact toric surface  X  which is semi-Fano (i.e., the anticanonical line bundleis nef). Unlike the Fano case, this involves nontrivial obstructions in the corresponding moduli problem. As a consequence, an explicit formula for the Lagrangian Floer superpotential  W  is obtained, which in turn gives an explicit presentation of the small quantum cohomology ring of  X . We also provide a computational verification of the conjectural ring isomorphism between the small quantum cohomology of  X  and the Jacobian ring of  W .

Open Gromov–Witten Invariants and Superpotentials for Semi-Fano Toric Surfaces