We provide an introduction to toroidal crossing spaces. These are a generalization of normal crossing spaces whose singularities are locally given by the boundary divisor of Gorenstein toric varieties. We focus on the sheaf of log smooth structures on a toroidal crossing space and its computation. Sections of this sheaf turn a toroidal crossing space into a generically log smooth family. This provides us with a means to construct generically log smooth families. In the case of normal crossing spaces, we have a very good understanding of the sheaf of log smooth structures.