Designed for a one year introductory course, this volume introduces students to the important techniques and theorems of Riemannian geometry, while presenting sufficient background on advanced topics to appeal to students who wish to specialize in the discipline. The text combines both the geometric parts of Riemannian geometry and the analytic aspects of the theory, and presents the most up-to-date research. The updated second edition includes such new material as: A completely new coordinate-free formula that is easily remembered, and is, in fact, the Koszul formula in disguise; an expanded number of coordinate calculations of connection and curvature; general fomulas for curvature on Lie Groups and submersions; variational calculus has been integrated into the text, which allows for an early treatment of the Sphere theorem using a forgotten proof by Berger; several recent results regarding manifolds with positive curvature.