This chapter is a first introduction to optimal vector quantization and its application to numerical probability. Optimal quantization produces the best approximation of probability distribution by finitely supported distributions in the sues of the Wasserstein distance. It naturally yields cubature formulas to compute expectations \(\mathbb {E}F(X)\) where X is an \(\mathbb {R}^{d}\)-valued random vector and F a Lipschitz or has a Lipschitz gradient (this applies too to conditional expectations). Thanks to A quantization based Richardson–Romberg extrapolations, tens e cubature formulas are shown to be competitive with regular Monte Carlo simulation at least up to 5 dimensions. A first approach to the computation of (quadratic) optimal quantizers of a given distribution is developed. Quantization is also investigated in Chap. 6 from an algorithmic view point and 11 as a numerical method to price Bermuda and American options.