In recent years, there has been much interest in the interplay between geometry and probability in high-dimensional spaces.  One striking result that has been established is a central limit theorem for random projections of random variables that are uniformly distributed in high-dimensional convex sets.   It is therefore natural to ask if such random projections also exhibit other properties satisfied by sums of iid random variables, such as large deviation principles.   As a step in this direction, we establish  (quenched and annealed) large deviation principles for random projections of sequences of random vectors that are uniformly distributed on $l^p$ balls in $n$-dimensional Euclidean space.   We also prove several interesting consequences of these principles, including the perhaps surprising fact that the well known Cramer’s theorem, which describes the large deviations of sums of iid random variables, is atypical.  Such questions, besides being of intrinsic interest, are also of relevance to statistics and data analysis.  This talk is based on joint work with Nina Gantert and Steven Kim.