A lot of standard notions known from Riemannian geometry have their counterparts for metric spaces. Busemann gave a definition of a nonpositively curved metric space using convexity properties of the distance function. It is also possible to study geodesics on a metric space. This book offers a systematic description of these points. In the first part, the author reviews basic notions about metric spaces (lengths of paths, length spaces and geodesic spaces, and distances). The second part of the book studies questions related to convexity in vector spaces. In the last part of the book, the author discusses Busemann spaces, locally convex metric spaces, their convexity properties and some further questions (including properties of convex functions, isometries, asymptotic rays and visual boundaries). An important role is played in the book by suitable examples, in particular by the Teichmüller space. Prerequisites needed to read the book are modest (basic facts on hyperbolic space and Teichmüller space). The book can be of interest for mathematicians working in analysis, geometry and topology.