In recent years Peng prososed a new notion called G-expectation, a type of nonlinear expectation motivated from dynamic risk measures with volatility uncertainty. On the other hand, a martingale under the G-expectation can be viewed as the solution to a "linear" Second Order Backward SDEs, the main subject of the short course which will be given by Nizar Touzi in this workshop. The theory has applications in many areas, e.g. Monte Carlo methods for fully nonlinear PDEs, finanancial problems in models with volatility uncertainty (volatility control, liquidity cost, Gamma constraint). Its main technical feature is the quasi-sure stochastic analysis, which invloves a class of mutually singular probability measures. In this talk we will introduce G-martingales, develop the quais-sure stochastic analysis, and establish the martingale epresentation theorem for G-martingales. This is a joint work with Mete Soner and Nizar Touzi.