<i>E</i>-polynomials of character varieties

This manuscript contains the basic ideas and constructions about e-polynomials in character varieties and the state of the art of certain research in the field, plus some new directions. We introduce mixed Hodge structures and E-polynomials, together with a series of arithmetic (counting points over finite fields) and geometric (stratification into parabolic types) techniques to compute them. We include a complete example of the calculation of the e-polynomial for the GL3-character variety of the free group. Finally, we extend the geometric stratification into parabolic types to a general reductive group G to obtain explicit motivic expressions for the G-character varieties, and reduce certain topological mirror symmetry conjectures for these moduli spaces.