M-Theory and Maximally Supersymmetric Gauge Theories

In this informal review for nonspecialists, we discuss the construction of maximally supersymmetric gauge theories that arise on the world-volume branes in string theory and M-theory. We focus on the relatively recent construction of M2-brane world-volume theories. In a formal sense, the existence of these quantum field theories can be viewed as predictions of M-theory. Their construction is therefore a reinforcement of the ideas underlying string theory and M-theory. We also briefly discuss the six-dimensional conformal field theory that is expected to arise on M5-branes. The construction of this theory is not only an important open problem for M-theory but also a significant challenge to our current understanding of quantum field theory more generally.