"This is a project in the area of Theoretical Computer Science, particularly discrete algorithms and computational complexity. Many Constraint Satisfaction Problems (`CSPs') such as Boolean satisfiability or graph coloring are well-known to be NP-hard, i.e., the worst-case computation time to solve these problems is exponential in the size of the problem instance. To illuminate the conceptual origins of the computational hardness of these problems, a major research effort over the past 30 years has been the study of Random instances of CSPs. Over the past decade, motivated by problems in statistical mechanics, physicists have developed stunningly detailed hypotheses on the structural and conceptual nature of random CSPs, based on ingenious but highly non-rigorous techniques. These hypotheses have led to a new class of Message Passing Algorithms, as well as to evidence that certain natural types of random CSPs may be computationally intractable. The goal of this project is to study these ideas rigorously and comprehensively from the perspective of the theory of computing."