We describe some new ideas and techniques introduced to study spaces with a given lower Ricci curvature bound, and discuss a number of recent results about such spaces.

Introduction In studying spaces with a given lower sectional curvature bound we have a very powerful tool in the Toponogov triangle comparison theorem. This allows us to study metric properties of such spaces (see for instance [Toponogov 1964; Burago et al. 1992; Perelman 1995]), and topological properties (see for instance [Cheeger 1991; Cheeger and Gromoll 1972; Grove and Shiohama 1977; Gromov 1981a; Grove and Petersen 1988; Perelman 1991]). Perhaps the most important tool for studying topological properties of such manifolds is the notion of critical points of distance functions in connection with the Toponogov triangle comparison theorem; see [Grove and Shiohama 1977] and compare with the remarks at the end of Section 1.

When we only assume a lower Ricci curvature bound, no such estimate is available. Classically, the only known general estimates of this type for Ricci curvature are the volume comparison theorem [Bishop and Crittenden 1964; Gromov 1981b] and the Abresch-Gromoll inequality [Abresch and Gromoll 1990].

In order to study manifolds with a given lower Ricci curvature bound there are at least two obstacles to overcome. First, many results from the sectional curvature case do not remain true for Ricci curvature. Second, due to the lack of a good estimate on the distance function we do not have good control on the local geometry in this case. We will see in this survey that in some sense the second obstacle is the most serious.