We present a review of the theory and applications of the Analytic Element Method as it exists today. The paper begins with a presentation of analytic elements used for modeling of divergence-free and irrotational flow in both two and three dimensions, including a description of the superblock approach, which makes it possible to deal effectively with very large models both in terms of accuracy and speed. We briefly discuss a particular application of the Analytic Element Method to multiaquifer problems, present the theory of the Analytic Element Method for general vector fields, and discuss the inclusion of source terms (irrotational vector fields with nonzero divergence). We finish the paper with a brief discussion of three major applications on a regional scale.