We analyze M5-instantons in F-theory, or equivalently D3-instantons with varying axio-dilaton, in the presence of 7-brane gauge groups. The chiral two-form on the M5-brane plays an important role, because it couples the M5-brane to vector multiplets and charged chiral fields. The chiral two-form does not have a semi-classical description. However if the worldvolume of the M5 admits a fibration over a curve with surface fibers, then we can reduce the worldvolume theory to an `MSW' CFT by shrinking the surface. For this class of MSW instantons, we can use heterotic methods to do computations. We explain this in some detail using the physical gauge approach. We further compare M5-instantons with D3-instantons in perturbative type IIb and find some striking differences. In particular, we show that instanton zero modes tend to disappear and constraints from chirality on instanton contributions to the superpotential evaporate for finite string coupling.