  We describe local mirror symmetry from a mathematical point of view and make
several A-model calculations using the mirror principle (localization). Our
results agree with B-model computations from solutions of Picard-Fuchs
differential equations constructed form the local geometry near a Fano surface
within a Calabi-Yau manifold. We interpret the Gromov-Witten-type numbers from
an enumerative point of view. We also describe the geometry of singular
surfaces and show how the local invariants of singular surfaces agree with the
smooth cases when they occur as complete intersections.
