This course is intended to be an approprite medium for developing the basic concepts and results of the Differential Calculus over normed linear spaces. The material is organized into 13 chapters as follows. Chapter 1 (with an introductory character) is devoted to the exposition of some algebraic and topological facts involving the normed linear spaces (linear continuous applications, equivalent norms, finite dimensional normed spaces). Chapters 2-7 are concerned with the classical questions of the Differential Calculus (in the precise framework): differentiable applications, directional derivatives, mean value theorems, local inversion and rank theorems, differentiable convex functions. In Chapter 8, an Integral Calculus theory is sketched (for the regulated functions). Chapters 9 and 10 are concerned with the Taylor development formula for differentiable maps and its applications to (constrained or not) extremum problems. Finally, Chapters 11-13 develop some questions belonging to the Differential Geometry over finite dimensional spaces (sub-manifolds, differential equations and differential forms). Any chapter of the course is accompanied with a lot of exercises, which have the role of giving a better understanding of its content. The exposition ends with a references list containing 19 titles and a subject index.