An Anosov representation of a hyperbolic group $\Gamma$ is a representation which quasi-isometrically embeds $\Gamma$ into a semisimple Lie group - say, SL(d, R) - in a way which generalizes and imitates the dynamical behavior of a convex cocompact group acting on a hyperbolic metric space. It is unknown whether every linear hyperbolic group admits an Anosov representation. In this talk, after motivating the theory of Anosov representations from the perspective of geometric group theory, I will discuss joint work with Sami Douba, Balthazar Flechelles, and Feng Zhu which shows that every hyperbolic group acting geometrically on a CAT(0) cube complex admits a 1-Anosov representation into SL(d, R) for some d. The proof exploits the relationship between the combinatorial/CAT(0) geometry of right-angled Coxeter groups and the projective geometry of a convex domain in real projective space on which a Coxeter group acts by reflections.