We study a quality-ladder model of endogenous growth that produces stochastic leadership cycles. Over a cycle, industry leaders can innovate several successive times in the same industry, gradually increasing the magnitude of their technological lead before being replaced by a new entrant. Initially, new leaders are eager to enlarge their lead and do much of the research, but if they innovate repeatedly, their propensity to invest in R&D decreases. Eventually they stop doing research altogether, and as they are overtaken a new cycle starts. The model generates a skewed firm size distribution and a deviation from Gibrat's law that accord with the empirical evidence. We also consider various policy measures, showing that in some cases policy should favour R&D by incumbents, not outsiders, and that stronger patent protection may reduce innovation and growth.