The standard mathematical approaches to topology, point-set topology and algebraic topology, treat points as the fundamental, unde ned entities, and construct extended spaces as sets of points with additional structure imposed on them. Point-set topology in particular generalises the concept of a`space' far beyond its intuitive meaning. Even algebraic topology, which concentrates on spaces built out of`cells' topologically equivalent to n-dimensional discs, concerns itself chie y with rather abstract reasoning concerning the association of algebraic structures with particular spaces, rather than the kind of topological reasoning which is required in everyday life, or which might illuminate the metaphorical use of topological concepts such as`connection' and`boundary'. This paper explores an alternative to these approaches, RCC theory, which takes extended spaces (`regions') rather than points as fundamental. A single relation, C (x; y) (read`Region x connects with region y') is axiomatised as the basis of the topological part of RCC theory. RCC theory has deep roots in the philosophical logic literature: advocacy of region-based rather than point-based spatial and temporal representation has links with a broader opposition to the view that set theory and predicate logic together provide an adequate basis for the formal representation of the world. The implications of adopting a region-based approach to topology such as RCC theory are discussed. RCC theory and conventional topology are compared as approaches to what might be called`commonsense' or`everyday' topology: the topological properties and relations of entities which can be embedded in a Euclidean space of three or fewer dimensions. The paper also covers the idea of`conceptual neighbourhoods', and their usefulness in reasoning about changes in spatial relations. Finally, some limitations, and possible applications, of RCC are discussed.