Gromov-Witten theory is the study of counts of curves in algebraic varieties. These counts may be formulated in several ways including log, open, orbifold and local variants. Recent years have seen the establishment of correspondences linking various curve counting invariants, often associated with different geometries. In this talk, my focus will be on the all genus correspondence linking log and local Gromov-Witten invariants of surfaces, and on specific examples. This is based on joint works with Andrea Brini and Pierrick Bousseau, and Navid Nabijou and Yannik Schüler.