In this thesis, we pursue a robust approach to pricing and hedging problems in mathematical finance. The general goal of this approach is to develop a pricing and hedging theory, which is based mainly on the market information than on a specific probabilistic belief about the future evolution of the risky assets. Motivated by the notion of prediction set in Mykland (2003), we include in our framework modelling beliefs through a set of paths to be considered, e.g. super-replication of a contingent claim is required only for paths falling in the given set. Our framework thus interpolates between model-independent and model-specific settings and allows quantifying the impact of making assumptions or gaining information. The first part of the thesis is concerned with robust fundamental theorem of asset pricing, pricing-hedging duality and their applications in a discrete-time setting in which some underlying assets and options, are available for dynamic trading and a further set of European options, possibly with varying maturities, is available for static trading. In the second part of the thesis, we consider the robust pricing-hedging duality problem with options in a continuous-time setting where underlying assets are assumed to have continuous paths. Our results include an "unconstrained" pricing-hedging duality, in the absence of options and beliefs, and a general but approximated pricing-hedging duality result. Moreover, when all put options are available for static hedging, the pricing problem is connected to the martingale optimal transport problem and our duality results in this thesis include the martingale optimal transport duality of Dolinsky and Soner (2013) and extend it to multiple maturities and multiple assets.