Riemannian geometry is characterized, and research is oriented towards and shaped by concepts (geodesics, connections, curvature, ...) and objectives, in particular to understand certain classes of (compact) Riemannian manifolds de?ned by curvature conditions (constant or positive or negative curvature, ...).Bywayofcontrast,geometricanalysisisaperhapssomewhatlesssyst- atic collection of techniques, for solving extremal problems naturally arising in geometry and for investigating and characterizing their solutions. It turns out that the two ?elds complement each other very well; geometric analysis o?ers tools for solving di?cult problems in geometry, and Riemannian ge- etry stimulates progress in geometric analysis by setting ambitious goals. It is the aim of this book to be a systematic and comprehensive int- duction to Riemannian geometry and a representative introduction to the methods of geometric analysis. It attempts a synthesis of geometric and - alytic methods in the study of Riemannian manifolds. The present work is the fourth edition of my textbook on Riemannian geometry and geometric analysis. It has developed on the basis of several graduate courses I taught at the Ruhr-University Bochum and the University of Leipzig. Besides several smaller additions, reorganizations, corrections (I am grateful to J.Weber and P.Hinow for useful comments), and a systematic bibliography,themainnewfeaturesofthepresenteditionareasystematic- troduction to K¨ ahler geometry and the presentation of additional techniques from geometric analysis.