We will give a mathematically-oriented review about the
geometry of the internal six-dimensional space M_6 in string theory
(with particular attention to the "type II" variety). In particular
we will be interested in vacua which have a property called
"supersymmetry." We will show what kind of constraints this
physical requirement puts on M_6. One reason this is interesting
mathematically is that the conditions we will get are a natural
generalization of the concept of Calabi-Yau manifold.