The number of all possible conformations of a polypeptide chain is too large to be sampled exhaustively. Nevertheless, protein sequences do fold into unique native states in seconds (the Levinthal paradox). To determine how the Levinthal paradox is resolved, we use a lattice Monte Carlo model in which the global minimum (native state) is known. The necessary and sufficient condition for folding in this model is that the native state be a pronounced global minimum on the potential surface. This guarantees thermodynamic stability of the native state at a temperature where the chain does not get trapped in local minima. Folding starts by a rapid collapse from a random-coil state to a random semi-compact globule. It then proceeds by a slow, rate-determining search through the semi-compact states to find a transition state from which the chain folds rapidly to the native state. The elements of the folding mechanism that lead to the resolution of the Levinthal paradox are the reduced number of conformations that need to be searched in the semi-compact globule (approximately 10(10) versus approximately 10(16) for the random coil) and the existence of many (approximately 10(3)) transition states. The results have evolutionary implications and suggest principles for the folding of real proteins.