Various recurrence and convergence results obtained in recent years indicate that dynamical systems exhibit regular behavior along polynomial times. In particular, weakly mixing systems turn out to always possess rather strong independence properties along certain sets of zero density. We will discuss some implications of these results in physics as well as applications to combinatorics and number theory (including polynomial extensions of Szemeredi's theorem on arithmetic progressions and recent work of Tao and Ziegler on polynomial patterns in primes). We will also formulate and discuss some natural open problems and conjectures.