There is growing evidence that we live in a negatively curved universe. A number of independent arguments suggest that the matter density is significantly less than the critical density: comparisions of density fluctuations and velocity fields (Willick & Strauss 1995; Davis, Nusser & Willick 1996; Riess et al. 1997); determination of mass-to-light ratios in clusters (Bahcall, Lubin & Dorman 1995); measurements of the baryon to dark matter ratio in clusters (S. White et al. 1993; Lubin et al. 1996; D. White, Jones & Forman 1997); the presence of more large scale structure than expected in flat models (Da Costa et al. 1994; Lin et al. 1996); the need to reconcile the value of the Hubble constant with globular cluster ages (Spergel, Bolte & Freedman 1997); the existence of large numbers of clusters at moderate redshift (Carlberg et al. 1997; Bahcall, Fan & Cen 1997). A number of groups have shown that CDM models with are compatible with microwave background measurements and large scale structure (Kamionkowski & Spergel 1994; Ratra et al. 1997). While we cannot rule out the possibility that there is a cosmological constant (e.g., Steinhardt & Ostriker 1995) that makes the universe flat, recent high redshift supernova observations (Perlmutter et al. 1997), as well as gravitational lens statistics (Turner 1990; Falco, Kochanek & Munoz 1997), suggest that this term is small.
There has been growing interest in the possibility that the universe is not only negatively curved, but compact (Gott 1980; Fagundes 1983; Cornish, Spergel & Starkman 1996a, 1996b; Bond, Pogosyan & Souradeep 1997; Levin et al. 1997). Our interest in the topology of the universe was stimulated by the possibility that it may be detectable and by the philosophical attractions of a finite universe (Cornish, Spergel & Starkman 1996a). This talk reviews some basic concepts in topology and then turns to the possibility of detecting the observational signature of a finite universe.