This book provides an introduction to Modern Riemannian Geometry, the geometry of curved metric spaces. Its main theme is the comparison of various spaces via various curvature conditions. Topics include

1. Calabi-Yau volume estimates
2. Gromov-Bishop volume comparison Theorem
3. New proofs of Rauch and Hessian comparison Theorem
4. Gromov's precompactness theorem
5. Cheeger finiteness theorem

These listed theorems are proved by using elementary differential inequalities. This is likely the first time to publish these interesting proofs in a book. This book is developed from lectures that the author gave at Cornell University, the University of Notre Dame and ``the 1998 Mathematics Summer School in China". Those lectures were very well received.

The book is suitable for first-year graduate students and upper level undergraduate students, whose specialties are not necessarily in differential geometry. The book can be used as a textbook for one semester courses.