The $G$-character variety of a surface of genus $g$ is the moduli space parametrizing representations of the fundamental group of the surface into $G$. Twisted character varieties consist 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 $G = \operatorname{SL}(2,\mathbb{C})$, $\operatorname{PGL}(2,\mathbb{C})$ and arbitrary genus. (joint work with V. Muñoz)