We prove that the scaling window of random k-SAT has width O(n^1/2+1/k), a polynomial improvement over our previous bound of O(n/ log n). Combined with a result of Abbe and Montanari, this establishes the satisfiability conjecture for every fixed k ≥ 3: there exists a constant αk such that a random k-SAT formula on n variables with ⌊αn⌋ clauses is satisfiable with high probability if α αk.