Summary: Saddlepoint approximations are derived for sums of independent, but not necessarily identically distributed random variables, along with generalizations to estimating equations and multivariate problems. These results are particularly useful for accurately approximating the distribution of regression coefficients. General formulas are given for the distribution of the coefficients arising out of a generalized linear model with both canonical and noncanonical link functions. We illustrate the case of logistic regression with a real data example and show how the Gibbs sampler may be used to obtain confidence sets for each regression parameter based on the saddlepoint approximation.