The method of moments is commonly used to reduce a kinetic equation into a fluid model. It can be seen as a semi-discretization with respect to the kinetic variable, and it results in a system of balance laws. However, such a system of moments is underdetermined and requires additional closure relations. The choices made in this derivation have impacts on the properties of the resulting system, namely the strong or weak hyperbolicity, the entropy dissipation, or the geometry of the admissible solution set.  In a first step, I will work on a kinetic toy problem, present its basic properties and the classical derivation of its system of moments. In a second step, I will give an overview of the different families of closures available in the literature, and discuss some recent advances for each of them. Eventually, I will reformulate this construction within a weak Galerkin framework and present first attempts to estimate the error produced by such approximations.