This article extends credibility theory by making quadratic adjustments that take into account the squared values of past observations. This approach amounts to introducing non-linearities in the framework, or to considering higher order cross moments in the com- putations. We first describe the full parametric approach and, for illustration, we examine the Poisson-gamma and Poisson-single Pareto cases. Then, we look at the non-parametric approach where premiums must be estimated based on data only, without postulating any type of distribution. Finally, we examine the semi-parametric approach where the con- ditional distribution is Poisson but the unconditional distribution is unknown. The goal of this paper is not to claim that q-credibility always brings better results than standard credibility, but it is to provide several building blocks for understanding how credibility changes when quadratic corrections are added.