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This thesis comprises various results that rely on the moments of a distribution or the sample moments associated with a set of observations. Since a sample of size n is uniquely specified by its first n moments, it is pertinent to make use of sample moments for modeling, classification or inference purposes. Three density mixtures are approximated by adjusting in various ways an initial density approximation referred to a base density by means certain moment-based functions, and the accuracy of the resulting density approximants are compared. A similar study is carried out in the context of density estimation. Moreover, it is explained that methodologies that are based on moments are, in fact, ideally suited to model massive data sets. Various types of quasi-Monte Carlo deterministic samples are then compared to randomly generated samples with respect to their distributional representativeness. As well, a novel methodology depending on an arctangent transformation is introduced for classifying the tail behaviour of probability laws. Finally, certain approximations to the distributions of quadratic forms in gamma, inverse Gaussian, binomial and Poisson random variables, which rely on a symbolic expansion of their moments, are proposed.