We say that a sequence a(n) is good for Szemer´edi’s theorem, if every subset of the integers with positive density contains arbitrarily long arithmetic progressions of the form m, m + a(n), ..., m + ka(n). A result of Bergelson and Leibman (1996) shows that if p is a polynomial with integer coefficients and zero constant term, then the sequence p(n) is good. We will give several new examples of good sequences that are not polynomial. To name a few, a(n) = [n √ 2008], b(n) = [n log n], or c(n) = [n 3/ log log n], are all good sequences. In fact, it turns out that if f(x) is a function that belongs to a Hardy field and satisfies some mild growth conditions, then the sequence [f(n)] is going to be good (joint work with M. Wierdl). For general sequences, it seems that certain equidistribution properties on nilmanifolds suffice to guarantee that a sequence is good. We will mention a related conjecture and a partial result (joint work with M. Wierdl and E. Lesigne).