  The mathematical features of a string theory compactification determine the
physics of the effective four-dimensional theory. For this reason,
understanding the mathematical structure of the possible compactification
spaces is of profound importance. It is well established that the
compactification space for M-Theory must be a seven-manifold with holonomy
$G_{2}$, but much else remains to be understood regarding how to achieve a
physically-realistic effective theory from such a compactification. Much also
remains unknown about the mathematics of these $G_{2}$-Manifolds, as they are
quite difficult to construct. This review discusses progress with regards to
both the mathematical and physical considerations surrounding spaces of
holonomy $G_{2}$. Special attention is given to the known constructions of
$G_{2}$-Manifolds and the physics of their corresponding M-Theory
compactifications.
