Perpetual futures—swap contracts that never expire—are the most popular derivative traded in cryptocurrency markets, with more than $100 billion traded daily. Perpetuals provide investors with leveraged exposure to cryptocurrencies, which does not require rollover or direct cryptocurrency holding. To keep the gap between perpetual futures and spot prices small, long position holders periodically pay short position holders a funding rate proportional to this gap. The funding rate incentivizes trades that tend to narrow the futures-spot gap. But unlike fixed-maturity futures, perpetuals are not guaranteed to converge to the spot price of their underlying asset at any time, and familiar no-arbitrage prices for perpetuals are not available, as the contracts have no expiry date to enforce arbitrage. Here, using a weaker notion of random-maturity arbitrage, we derive no-arbitrage prices for perpetual futures in frictionless markets and no-arbitrage bounds for markets with trading costs. These no-arbitrage prices provide a valuable benchmark for perpetual futures and simultaneously prescribe a strategy to exploit divergence from these fundamental values. Empirically, we find that deviations of crypto perpetual futures from no-arbitrage prices are considerably larger than those documented in traditional currency markets. These deviations comove across cryptocurrencies and diminish over time as crypto markets develop and become more efficient. A simple trading strategy generates large Sharpe ratios even for investors paying the highest trading costs on Binance, which is currently the largest crypto exchange by volume.