Quantum field theorists often resort to perturbative expansions in a reasonably small parameter. Yet these may have no radius of convergence. Summation of sign-alternating asymptotic series is sometimes achievable by Borel transformation, which may be improved by Pade approximations. I shall give an example that yields to these methods. The less tractable sign-constant case occurs in condensed-matter problems. Here, Michael Borinsky and I have developed a trans-series. This is a formal expansion in powers of x, exp(-1/x) and log(x), with coefficients whose asymptotic growth is determined by further terms in the trans-series, and so on, ad infinitum.