Elections rely on people, hardware, and software, all of which are fallible and subject to manipulation. Voting equipment is built by private vendors using foreign parts. Many states outsource election results reporting. Advanced persistent threats and insider threats are real. Uncertainty about the outcome of elections has been weaponized politically recently. We need to conduct elections in a way that provides affirmative evidence that the reported winners really won–despite malfunctions, errors, and malfeasance. Evidence-based elections require voter-verified (generally, hand-marked) paper ballots kept demonstrably secure throughout the canvass and manual audits of election results against the trustworthy paper trail. Compliance audits establish whether the paper trail is complete and trustworthy. Risk-limiting audits (RLAs) check the outcome by testing the hypothesis that one or more reported winners did not win. For a broad variety of social choice functions, including plurality, multi-winner plurality, supermajority, Borda count, approval voting, all scoring rules, instant-runoff voting (ranked-choice voting), and D'Hondt and Hamiltonian proportional representation, the hypothesis that at least one reported winner did not win can be reduced to the hypothesis that the mean of one or more lists of nonnegative numbers is not greater than 1/2. Martingale tests of these nonparametric hypotheses sequentially are especially practical. Methods to accommodate different sampling plans, equipment capability, logistical constraints, and laws and regulations have been developed and piloted in more than a dozen states in jurisdictions of all sizes, including roughly 10 audits of statewide contests. RLAs are in law in several states.