The notion of sublinear expectation (also called upper expectation, coherent pro vision in statistics) of which the well-known relation (E)[X] + (E)[Y] =(E)[X + Y] in clas sical probabilitytheory becomes the following sublinear one (E)[X] + (E)[Y] ≤ (E)[X +Y],is proved to be a basic toolin decision making under mean-uncertainty variance uncertainty which is crucial in superhedging, superpricing and measures of risk in finance.We will discussfrom an point view of statistics and probabilities our new results of the law of large number and central limit theorem under a space of sublin ear expectation (Ω,F, (E)), instead of the well-known framework of probability space (Ω, F,P) introduced by Kolmogorov.The notion of G-normal distribution plays a central role.This is also the start point a new theory of random and stochastic calculus which gives us a new insight to characterize and calculate varies kinds of financial risk.A G-Brownian motion Bt, t ≥ 0,is a path-wise continuous stochastic process defined in a sublinear expectation space (in the place of probability space) with stationary and independent increments.Its quadratic variation process t, t ≥ 0 is also a continuous process with stationary and independent increments.We present the related random and stochastic calculus and their applications to option pricing to measures of risk in finance under drift and volatility uncertainty.