In this thesis, we study motivic classes of punctual Hilbert schemes and their relationship to Igusa zeta functions and motivic integration.

We start with the case of varieties in a smooth ambient variety, for which we provide an equality between the generating series of motivic classes of curvilinear punctual Hilbert schemes with a particular Igusa zeta function, that we refer to as the curvilinear Igusa zeta function. Studying the behavior of arcs under embedded resolutions, we can express the curvilinear Igusa zeta function, hence the motivic class of the curvilinear open, in terms of an embedded resolution of singularities of the given variety. In the case of plane curves, this expression implies the topological invariance of the motivic class of the curvilinear open, together with the polynomiality of the class with respect to the variable given by the motivic class of the affine line in the Grothendieck ring of varieties.

Still in the case of plane curves, we also establish an equality between the generating series of motivic classes of unibranch punctual Hilbert schemes and the classic Igusa zeta function, utilizing the formal setting provided by the space of branches. Here, too, we describe the motivic class of the unibranch open in terms of an embedded resolution of singularities of the given curve, and we find its topological invariance and its polynomiality in the Grothendieck ring.

Finally, we focus on the case of unibranch plane curves and generalize our work to the entire punctual Hilbert scheme. We prove the existence and uniqueness of some special representatives for the points of the punctual Hilbert scheme, from which we obtain an explicit formulation of the motivic class of the fixed-generators punctual Hilbert scheme in terms of certain subvarieties of determinantal varieties arising from the syzygies of the given curve. Specializing these results to the case of $(p,q)$-curves, we find an explicit connection between the fixed-generator locus and the one-generator locus of the punctual Hilbert scheme, leading to the polynomiality in the Grothendieck ring with positive coefficients and the topological invariance of the motivic class of the punctual Hilbert scheme.