Abstract: A Q-acyclic complex is a higher-dimensional analogue of a tree. We study a natural model of random 2-dimensional Q-acyclic complex, following earlier enumerative work of Kalai and probabilistic work of Lyons. We are especially interested in the expected topological properties. We show that, asymptotically almost surely, a random 2-dimensional Q-acyclic complex is aspherical, i.e. has a contractible universal cover. We also show that the torsion in homology grows exponentially fast in the number of faces, and that the fundamental group is hyperbolic in the sense of Gromov. We will also discuss some open problems. In particular, the fundamental group of a random 2-dimensional Q-acyclic complex seems to be a candidate for a one-ended hyperbolic group without surface subgroups. This talk is based on joint work with Andrew Newman.