Many applications in financial economics require calculation of conditional moments or contingent claims prices, but such expressions are known in closed-form for only a few specific models. We develop a method for approximation of such quantities, using power series, for a large class of non-affine diffusions and interest rate specifications. We also introduce a family of non-affine transformations of time that often dramatically improve the convergence properties of the power series approximations. In many cases, the approximations are uniformly accurate for arbitrarily long time horizons, and are therefore suitable for applications such as bond pricing, in which the time-to-maturity may be many years. The ability to approximate solutions accurately and in closed-form simplifies the estimation of non-affine continuous-time term structure models, since the bond pricing problem must be solved for many different parameter vectors during a typical estimation procedure. We show through a bond pricing example that the approximations are easy to derive and highly accurate over a wide range of interest rate levels for arbitrarily long maturities.