  This PhD Thesis is devoted to the study of Hodge structures on a special type
of complex algebraic varieties, the so-called character varieties. For this
purpose, we propose to use a powerful algebro-geometric tool coming from
theoretical physics, known as Topological Quantum Field Theory (TQFT). With
this idea in mind, in the present Thesis we develop a formalism that allows us
to construct TQFTs from two simpler pieces of data: a field theory (geometric
data) and a quantisation (algebraic data). As an application, we construct a
TQFT computing Hodge structures on representation varieties and we use it for
computing explicity the Deligne-Hodge polynomials of parabolic
$\mathrm{SL}_2(\mathbb{C})$-character varieties.
