In this project our goal is to deal with a class of hedging and pricing problems whicharise in modern Mathematical Finance.These problems are not only interesting fromthe applications point of viewbut also provide a good source for new mathematical questions which requirenew tools in the area of probability theory.We will focus on three main topics:(i) Hedging with Friction.(ii) Robust Hedging.(iii) Numerical Schemes.All the above topics are related to the theory ofpricing and hedging of derivative securities.In the last 35 years there was great progress in this direction.By now there is quite a good understanding of pricing and hedgingin frictionless financial markets with a known probabilistic structure.This understanding was achieved by developing the machinery of stochastic calculus,stochastic control, martingale theory,hypothesis testing, etc.In real market conditions,it is very difficult to providea correct probabilistic model for the behaviorof stock prices.Furthermore, trading of assets is subject to transaction costs,i.e. there is a gap between an ask price and the bid price.These two factsraise the natural questionof understanding hedging in markets with friction and model uncertainty.Usually, when dealing with complex models of financial markets, explicitformulas for option prices and the corresponding super--replication strategiesare not available. This is the motivation to study numerical schemesfor several stochastic control problems which are related to hedging under volatility uncertainty.In the current project we are interested not only in providing the algorithms of numerical schemes,but also in implementing them.In summary,the proposed questions are not only crucial for the understandingof pricing andhedging in financial markets, but also a great source of new mathematical problems.These problems require new tools and also attract the attention of world class mathematicians.