In all of biology, there is no more consequential problem than the origin of life; yet despite the expenditure of much effort and ink over the past seventy years a satisfactory answer continues to elude us. This chapter surveys the huge body of experimental and theoretical work on this problem. Topics include the primordial broth of abiotic chemicals, prebiotic synthesis of metabolites, the quest for a self-replicating ribozyme, the critical importance of energy coupling, self-organized metabolic cycles, why membranes are essential, and the importance of natural selection from the outset. The recent idea, that life arose in the interstices of mineral deposits at the margins of warm alkaline hydrothermal vents, receives special attention. Between the first proto-cells and the Last Universal Common Ancestor, all the machinery of life must have been invented. We consider the place of the RNA World, and the horrendous difficulty of envisaging the origin of translation. In the end, the origin of life remains a mystery that passes understanding.