One of the most challenging problems in probability and complexity theory is to establish and determine the satisfiability threshold, or phase transition, for random k-SAT instances: Boolean formulas consisting of clauses with exactly k literals. As the previous part of the volume has explored, empirical observations suggest that there exists a critical ratio of the number of clauses to the number of variables, such that almost all randomly generated formulas with a higher ratio are unsatisfiable while almost all randomly generated formulas with a lower ratio are satisfiable. The statement that such a crossover point really exists is called the satisfiability threshold conjecture. Experiments hint at such a direction, but as far as theoretical work is concerned, progress has been difficult. In an important advance, Friedgut [23] showed that the phase transition is a sharp one, though without proving that it takes place at a “fixed” ratio for large formulas. Otherwise, rigorous proofs have focused on providing successively better upper and lower bounds for the value of the (conjectured) threshold. In this chapter, our