We study nonatomic, locally positive, Lebesgue-Stieltjes measures on compact Menger manifolds and show that the set of all ergodic homeomorphisms on any compact Menger manifold X forms a dense G δ set in the space of all measure preserving autohomeomorphisms of X with the compact-open topology. In particular, there exists a topologically transitive homeomorphism on any compact Menger manifold, which answers a question posed by several authors.We also prove the existence of homeomorphisms that are chaotic in the sense of Devaney as well as everywhere chaotic in the sense of Li-Yorke.