Von Neumann's Mean Ergodic Theorem and Birkhoff's Pointwise Ergodic Theorem lie in the foundation of Ergodic Theory. Over the years there have been many generalizations of the two, most recently a version of pointwise ergodic theorem for measure-preserving actions of amenable groups due to Elon Lindenstrauss. In the first chapter, we extend some of Lindenstrauss' results to measure-preserving actions of countable left-cancellative amenable semigroups and to averaging along more general types of Folner sequences. In the next three chapters, we study convergence of Cesaro averages of a special form for measure-preserving actions of countable amenable groups. We extend some of the results obtained by D.Berend and V.Bergelson for joint properties of Z-actions to joint properties of actions of countable amenable groups. In particular, we obtain a criterion for joint ergodicity of actions of countable amenable groups by automorphisms of a not necessarily abelian compact group.
In the last chapter of this dissertation, we investigate convergence of Cesaro averages for two non-commuting measure-preserving transformations along a regular sequence of intervals.