This monograph concerns the smooth classification of a certain class of algebraic surfaces, namely regular elliptic surfaces of geometric genus one, (elliptic surfaces with bl = 0 and b2+ = 3). The authors give a complete classification of these surfaces up to diffeomorphism. They achieve this result by partially computing one of Donaldson’s polynomial invariants. The computation is carried out using techniques from algebraic geometry. In these computations, both the basic facts about the Donaldson invariants and the relationship of the moduli space of ASD connections with the moduli space of stable bundles are assumed known. Some familiarity with the basic facts of the theory of moduli of sheaves and bundles on a surface is also assumed.