Modern electron theory of metals abounds in geometric terminology and, by using geometric ideas, provides clear descriptions of many complicated phenomena. Some problems in the theory of normal metals that can be interpreted geometrically are examined in this paper in a language accessible to many physicists. Particular attention is devoted to those phenomena and properties that are connected with qualitative (topological) changes in geometric figures such as Fermi surfaces, plane sections though such surfaces, ``belts'' on the Fermi surface, and so on. Graphical illustrations of these ideas are provided.