In this talk, I will discuss a model for complex networks that was introduced recently by Krioukov et al. In this model, N points are chosen randomly on the hyperbolic plane and any two of them are joined by an edge if they are within a certain hyperbolic distance. The model turns out to behave similarly to the well-known Chung-Lu model, but without the independence between the edges. In particular, it exhibits a power-law degree sequence, small distances and, unlike the Chung-Lu model and many other well-known models for complex networks, it also exhibits clustering. I will present some results on the component structure of the graph model, and on the probability that it is connected. (based on joint work with Michel Bode and Nikolaos Fountoulakis)