This book is about the smooth classification of a certainclass of algebraicsurfaces, namely regular ellipticsurfaces of geometric genus one, i.e. elliptic surfaces withb1 = 0 and b2+ = 3. The authors give a completeclassification of these surfaces up to diffeomorphism. Theyachieve this result by partially computing one of Donalson'spolynomial invariants. The computation is carried out usingtechniques from algebraic geometry. In these computationsboth thebasic facts about the Donaldson invariants and therelationship of the moduli space of ASD connections with themoduli space of stable bundles...