In this paper we investigate the ownership and control of British firms using recent techniques from computational graph theory. In particular, we analyze the `small-world' properties of UK company ownership and the corporate elite. A `small-world' is characterized by short `path-lengths' (actors are linked by a short chain of acquaintances) and high `clustering' (one's friends tend to be friends in their own right). We find that the network of both ownership and control can be characterized as a small-world. We simulate a set of corporate worlds using newly introduced random-graph models of Chung and Lu. In general, we reject the hypothesis that the corporate world of ownership and control is generated by a random graph model. The network structure of ownership and boards is decidedly more `clubby' than would be expected by chance, suggesting the presence of additional social structure not captured by the random graph model. In addition, we find that financial institutions are important and give rise to different network topologies.