Let M be a connected, finite dimensional, second countable manifold and let µo be a locally finite, "а-good", positive Borel measure on M. Let Mc(M) be the group of all compactly supported homeomorphisms of M, and let Hc(M, µo) be the group of all measure preserving, compactly supported homeomorphisms of M. These groups are given the so called direct limit topology. The purpose of these thesis is to prove the following results. Theorem. The group Hc(M, µo) is locally contractible (see 4.9). Theorem. The inclusion H (M, µo) «-► Hc (M) is a weak homotopy equivalence (see 4.11). Remark. Similar results hold for homeomorphisms fixing the boundary of M pointwise. Theorem. Let M be a connected, second countable manifold without boundary and of dimension n ≥ 3, and let µo be a "а-good" measure on M. Let Hc,d (M, µo) be the path component of the identity in HC(M. u0). Then the abellanization of Hc,o (M, µo) is isomorphic to a quotient of the first real homology group H | (M, IR) of M by some discrete subgroup Г. The group Г vanishes whenever M is non-compact. The commutator subgroup of Hc,o (M, µo) is simple and it is generated by all those elements in Hc,o (M, µo) which are supported in topological n-balls (see 8.14 and 8.16).