On string compactifications to two (and three) dimensions: Critical and noncritical

This thesis is devoted to certain aspects of string compactifications to two and three dimensions. We construct vacua that describe the propagation of non-critical strings on a singular Spin(7) space. We also examine compactifications which preserve six supercharges, and obtain the classical and quantum moduli space of such theories. We also consider compactifications on T8 / Z2 and point out the relevance of these flux compactifications to understanding the mathematical result of Guan and Beauville on the Hodge numbers of hyper-Kahler four-folds.