Market impact is the response of prices to trades and is a fundamental quantity to understand how supply and demand affect price, but also an important component of transaction costs. In this talk, I introduce the still open problem of mathematical modeling of market impact in a way which is consistent with data but also lacking dynamical arbitrage opportunities. I review some models proposed in the mathematical finance literature and discuss their comparison with data, deriving necessary conditions for the absence of dynamical arbitrage. I then focus on the optimal execution problem in continuous and discrete time, deriving the solution under different specifications of the impact model and of the chosen benchmark.