In search of infinite-dimensional Kahler geometry

This paper is devoted to a survey of recent results in the Kahler geometry of infinite-dimensional Kahler manifolds. Three particular classes of such manifolds are investigated: the loop spaces of compact Lie groups, Hilbert-Schmidt Grassmannians, and the universal Teichmuller space. These investigations have been prompted both by requirements in Kahler geometry itself and by connections with string theory, which are considered in the last section. Bibliography: 43 titles.