In these notes, we first introduce the theory of arbitrage and pricing for frictionless models, i.e. the classical theory of mathematical finance. The main classical results are presented, i.e. the characterization of absence of arbitrage opportunities, based on convex duality, and dual characterizations of super-hedging prices are deduced. We then present financial market models with proportional transaction costs. We discuss no arbitrage conditions and characterize super-hedging prices as in the frictionless case. Another approach based on the liquidation value concept is finally introduced.