This book describes the category of metric manifolds and metric maps to which a broad class of theorems and constructions extend from the realm of compact manifolds. The category is a broad one because all paracompact manifolds admit metric structures. Metric theorems include compact theorems because a compact manifold admits a unique metric structure and with respect to it any smooth map with compact domain is a metric map. Finally, there is a sufficient abundance of metric maps that, for example, structural stability under perturbation within the family of metric maps remains useful.