This monograph presents an approach to the measure-theoretical foundations of statistics and the theory of sufficiency, covering undominated and dominated statistical experiments. The familiar topics in the dominated case, such as pairwise sufficiency, Neyman factorization, minimal sufficient statistics, and the Rao-Blackwell theorem, are treated from a more general viewpoint than in the Halmos-Savage-Bahadur scheme and sometimes in a slightly different way. The main theme is that if the usual notion of sufficiency in terms of conditional probabilities is modified to omit the standard assumption of a common dominating sigma-finite measure, then certain aspects of statistics become more straightforward. In particular, one can extend Neyman's factorization criterion. Consequently, this will be of interest to researchers in statistics and may even lead to new developments in the theory of sufficiency.