This article describes approximations to the posterior means and variances of positive functions of a real or vector-valued parameter, and to the marginal posterior densities of arbitrary (i.e., not necessarily positive) parameters. These approximations can also be used to compute approximate predictive densities. To apply the proposed method, one only needs to be able to maximize slightly modified likelihood functions and to evaluate the observed information at the maxima. Nevertheless, the resulting approximations are generally as accurate and in some cases more accurate than approximations based on third-order expansions of the likelihood and requiring the evaluation of third derivatives. The approximate marginal posterior densities behave very much like saddle-point approximations for sampling distributions. The principal regularity condition required is that the likelihood times prior be unimodal.