We integrate heterogeneity and uncertainty in investor valuations into a model of takeovers. Investors have dispersed valuations, holding shares in firms they value more highly, and a successful offer must win approval from the median target shareholder. We derive the consequences for an acquiring firm’s takeover offer—its size and cash/equity structure—and implications for takeover premia, firm returns, share price dynamics, the probability that a takeover succeeds, and shareholder welfare. We characterize when the acquirer prefers cash offers and when equity offers are best. Our model collectively reconciles various empirical regularities that have proven elusive to explain in one unified framework, and we derive new testable predictions. This paper was accepted by Neng Wang, finance.