This text offers a sampling of the expanding field of artificial neural network theory. It considers select areas of discrete mathematics linking combinatorics and the theory of the simplest types of artificial neural networks. Neural networks have emerged as a key technology in many fields of application, and an understanding of the theories concerning what such systems can and cannot do is essential. The author discusses connections between special types of Boolean functions and the simplest types of neural networks. Some classical results are presented with accessible proofs, together with some more recent perspectives, such as those obtained by considering decision lists. In addition, probablistic models of neural network learning are also discussed and graph theory, some partially ordered set theory, computational complexity, and discrete probability are among the mathematical topics involved.