With a focus on statistical inference, distributional and statistical properties of multivariate and multiparameter exponential families are studied, such as generating functions, marginal and conditional distributions, and product measures. Sufficiency and completeness of statistics in exponential families is examined, and representations of the score statistic and the Fisher information matrix are shown. Many examples illustrate the results. Several divergence and distance measures are considered, and representations in terms of the cumulant function and the mean value function are given.