G$_{2}$-Manifolds and M-Theory Compactifications

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.