A classical theorem due to H. Weyl states that if P is a real polynomial such that at least one of its coefficients (other than the constant term) is irrational, then the sequence P(n), n=1,2,… is uniformly distributed mod 1. After briefly reviewing various approaches to the proof of Weyl’s theorem, we will discuss some recent extensions which involve “generalized polynomials”, that is, functions which are obtained from the conventional polynomials by the use of the greatest integer function, addition and multiplication. We will explain the role of dynamical systems on nil-manifolds in obtaining these results and discuss the intrinsic connection between the generalized polynomials and the polynomial extensions of Szemeredi’s theorem on arithmetic progressions. We will also discuss the recent joint work with Leibman on characterization of nil-translations in terms of recurrence. We will conclude with formulating and discussing some natural open problems and conjectures.