Abstract: The G-character variety of a surface of genus g is the moduli space parametrizing repre- sentations of the fundamental group of the surface into G. Twisted character varieties con- sist of representations of the fundamental group of a punctured surface with fixed holonomy around the puncture. Using a geometric technique based on stratifications and fibrations of the moduli space, we will show how to compute the E-polynomials of these spaces for G = SL(2,C), PGL(2,C) and arbitrary genus. (joint work with V.Muñoz)