  A self-contained review is given of the matrix model of M-theory. The
introductory part of the review is intended to be accessible to the general
reader. M-theory is an eleven-dimensional quantum theory of gravity which is
believed to underlie all superstring theories. This is the only candidate at
present for a theory of fundamental physics which reconciles gravity and
quantum field theory in a potentially realistic fashion. Evidence for the
existence of M-theory is still only circumstantial---no complete
background-independent formulation of the theory yet exists. Matrix theory was
first developed as a regularized theory of a supersymmetric quantum membrane.
More recently, the theory appeared in a different guise as the discrete
light-cone quantization of M-theory in flat space. These two approaches to
matrix theory are described in detail and compared. It is shown that matrix
theory is a well-defined quantum theory which reduces to a supersymmetric
theory of gravity at low energies. Although the fundamental degrees of freedom
of matrix theory are essentially pointlike, it is shown that higher-dimensional
fluctuating objects (branes) arise through the nonabelian structure of the
matrix degrees of freedom. The problem of formulating matrix theory in a
general space-time background is discussed, and the connections between matrix
theory and other related models are reviewed.
