The authors present a survey of Riemannian geometry that sketches the main developments in that subject through about 1985; no bibliographic references after that date exist. They begin with historical remarks and brief descriptions of the contributions of Lobachevski, Gauss, Riemann, F. Klein, E. Cartan, Ricci and Levi-Civita. The article consists of four chapters, and the main themes considered are geodesic behavior, the relationship between geometry and topology and properties of spaces with additional structure (symmetric, homogeneous, Kähler, etc.). The geometric methods emphasized include the variational theory of geodesics and the properties of the Laplace operator on functions. Topics omitted, partly by decision of the authors and partly by the 1985 constraint, include the Chern-Weil theory of characteristic classes, geodesic flows on Riemannian manifolds with negative sectional curvature, geodesic flows in the context of Hamiltonian systems (especially complete integrability), geometric group theory and the theory of Alexandrov spaces as extended by Gromov and others. Important articles are occasionally mentioned in the text but not in the bibliographic references (e.g. the splitting theorem of Cheeger-Gromoll for manifolds with nonnegative Ricci curvature). This article is a valuable resource for geometers and others who want to become acquainted with the important results of Riemannian geometry through the early to middle 1980's. Definitions and examples of important geometric objects and concepts are included, but the description is brief and not intended to be a text for those who want to learn topics from the beginning. We give a brief summary of the contents of the four chapters. Chapter 1 is devoted to a discussion of basic concepts of Riemannian geometry including affine and Riemannian connections, parallel translation and holonomy, curvature, isometries and homogeneous spaces. Chapter 2 discusses geodesic behavior. Geodesics are presented as critical points of the energy functional, and the index form of a geodesic is introduced. Particular topics include Jacobi vector fields, conjugate points, the Morse index theorem, comparison theorems of Rauch, Morse, Toponogov and existence problems and results for closed geodesics. The third chapter is the longest, and its theme are the relationships between geometry and topology, primarily for manifolds whose sectional, Ricci or scalar curvature has a fixed sign. For positive or nonnegative curvature the topics discussed include the sphere theorems of Berger, Klingenberg, Grove-Shiohama and others for compact manifolds and the theorems of Cheeger, Gromoll-Meyer, Toponogov and Cheeger-Gromoll for noncompact manifolds. For manifolds with negative sectional curvature there is a discussion of the relationship between the algebraic properties of the fundamental group and the geometry of the manifold. Specific topics include the Cartan-Hadamard theorem, the flat torus theorem and the splitting theorem of Gromoll-Wolf and Lawson-Yau. Chapter 3 also contains a discussion of some important contributions of Gromov, including almost flat manifolds, simplicial volume and fill in radius. This chapter has a separate section on the Ricci tensor that includes a discussion of the diameter theorem of Myers (but not the sharp extension by S. Cheng) and the problem of finding Riemannian manifolds with prescribed Ricci tensor. Chapter 3 concludes with a discussion of the problem of finding Riemannian metrics with scalar curvature of a fixed sign. Chapter 4 deals primarily with Riemannian manifolds that have additional structure such as symmetric spaces, homogeneous spaces or Kähler manifolds. Topics include the Mostow rigidity theorem, the Calabi conjecture and the existence of Einstein metrics. The chapter concludes with a discussion of pseudo-Riemannian manifolds, classical mechanics from the viewpoint of Riemannian geometry and the theory of Yang-Mills connections.