I will start with recalling few highlights of Furstenberg's ergodic theoretic proof of the Szemeredi theorem on arithmetic progressions and will exhibit a new extension of the latter. The proofs are related to ergodic theorems for nonconventional sums which, from probabilistic viewpoint, are akin to studying laws of large numbers for such sums. This motivated my interest to o btain other classical limit theorems of probability theory for nonconventional sums which will be discussed in the main part of my talk and I'll do my best to make it accessible to the general mathematical audience.