In this chapter we discuss the basic theory of Gaussian quadrature, which is at the heart of quadrature-based moment methods (QBMM). Proofs for most of the results are not included and, for readers requiring more extensive analytical treatments, references to the literature are made. In addition to a summary of the relevant theory, different algorithms to calculate the abscissas (or nodes) and the weights of the quadrature approximation from a known set of moments are presented, and their advantages and disadvantages are critically discussed. It is important to remind readers that most of the theory for quadrature formulas was developed for mono-dimensional integrals, namely integrals of a single independent variable. Therefore the discussion below starts from univariate distributions, for which the Gaussian quadrature theory applies exactly, and subsequently moves to bivariate and multivariate distributions. Although in the latter cases the quadrature is no longer strictly Gaussian, most of its interesting properties are still valid. In the univariate case, the weights and abscissas are used in the quadrature method of moments (QMOM) to solve moment-transport equations. Thus, we will refer to moment-inversion algorithms that use a full set of moments as QMOM, while other methods that use a reduced set will be referred to differently.