A new approach to N = 2 supersymmetry based on the concept of harmonic superspace is proposed and is used to give an unconstrained superfield geometric description of N = 2 super Yang-Mills and supergravity theories as well as of matter N = 2 hypermultiplets. The harmonic N = 2 superspace has as independent coordinates, in addition to the usual ones, the isospinor harmonics u : on the sphere SU(2)/U(l). The role of u : is to relate the SU(2) group realised on the component fields to a U ( l ) group acting on the relevant superfields. Their introduction makes it possible to SU(2)-covariantise the notion of Grassmann analyticity. Crucial for our construction is the existence of an analytic subspace of the general harmonic N = 2 superspace. The hypermultiplet superfields and the true prepotentials (pre-prepotentials) of N = 2 super Yang-Mills and supergravity are unconstrained superfunctions over this analytic subspace. The pre-prepotentials have a clear geometric interpretation as gauge connections with respect to the internal SU(2)/U( 1) directions. A radically new feature arises: the number of gauge and auxiliary degrees of freedom becomes infinite while the number of physical degrees of freedom remains finite. Other new results are the massive N = 2 Yang-Mills theory and various off-shell selfinteractions of hypermultiplets. The propagators for matter and Yang-Mills superfields are given.

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