This book originated from a course of lectures given at Yale University during 1968-69 and a more elaborate one, the next year, at the Tata Institute of Fundamental Research. Its aim is to present a detailed ac- count of some of the recent work on the geometric aspects of the theory of discrete subgroups of Lie groups. Our interest, by and large, is in a special class of discrete subgroups of Lie groups, viz., lattices (by a lattice in a locally compact group G, we mean a discrete subgroup H such that the homogeneous space GJ H carries a finite G-invariant measure). It is assumed that the...