I show that non-linear pricing of market risk can explain many prominent cross-sectional stock return anomalies, such as momentum, betting-against-beta, idiosyncratic volatility, and liquidity. The non-linear pricing model is inferred from options data without assumptions on the pricing relationship. I further document that many anomalies have a strong tail risk exposure, which is successfully priced by the model. A key feature of the model is a sizable upside risk premium of approximately 4% per annum. Finally, the pricing results can be explained with a compensation for exposure to systematic variance risk.