The aim is to show how to deal with model uncertainty in finance, without imposing the no-arbitrage condition. The idea is to quantify the notion of arbitrage, and obtain a quantitative version of the Fundamental Theorem of Asset Pricing and of the Super-Replication Theorem. As a consequence we can formalize the following statements: in a market that admits “small arbitrage” the “pricing measures” are such that asset price process is “close to being a martingale”, or equivalently, that hedging strategies need to cover some additional “small costs”. Finally, we study robustness of the amount of arbitrage and existence of respective pricing measures, showing stability of these concepts with respect to an L-infinity version of the adapted Wasserstein distance. Based on joint work with B. Acciaio and G. Pammer.