abstract: Properly convex, real projective manifolds form an interesting class of aspherical manifolds, including the hyperbolic ones. The topology of such (closed, orientable) manifolds is clear in dimension 2 and partially understood in dimension 3, but little is known in higher dimensions. The goals of the talk are to give a partial overview on what is known from this viewpoint, and to shed some light on dimension 4. Based on a work in progress with A. Seppi and L. Slavich.