This book studies a new theory of metric geometry on metric measure spaces. The theory was originally developed by M. Gromov in his book Metric Structures for Riemannian and Non-Riemannian Spaces and based on the idea of the concentration of measure phenomenon by Levy and Milman. A central theme in this book is the study of the observable distance between metric measure spaces, defined by the difference between 1-Lipschitz functions on one space and those on the other. The topology on the set of metric measure spaces induced by the observable distance function is weaker than the measured Gromov-Hausdorff topology and allows the author to investigate a sequence of Riemannian manifolds with unbounded dimensions. One of the main parts of this presentation is the discussion of a natural compactification of the completion of the space of metric measure spaces. The stability of the curvature-dimension condition is also discussed.