In this talk I'll survey results concerning the normalized Ricci flow evolving from a conformally compact asymptotically hyperbolic metric. I'll discuss joint work with Mazzeo and Woolgar on the behavior of the renormalized volume along the flow of both asymptotically Poincaré-Einstein metrics and metrics with an even expansion. I'll then discuss joint work with Woolgar on the long-time existence of the flow for rotationally symmetric asymptotically hyperbolic initial data, and ongoing work with Guenther and Isenberg on the stability of these flows.