The book is intended for students in their third year of a bachelor degree or in their first year of a master degree in mathematics, and it is divided into two parts. The first one, which is structured in six chapters, is devoted to the elements of a course on variational analysis and optimization, and in the second part an interesting collection of 103 solved exercises and problems, classified in order of difficulty, is presented. These exercises and problems are organized in a progressive form, firstly those in finite-dimensional spaces, and later the ones in infinite-dimensional spaces (see the summary of the contents below). In my opinion, the second part of the book is the most original and interesting one, because it allows the reader to familiarize himself with the concepts and theoretical results of the course. The authors show a collection of very motivating problems, inciting the reader to solve them. The book is carefully written and will undoubtedly serve as a useful resource not only for advanced undergraduates and graduate students, but also for teachers, professional mathematicians and researchers in several fields. I. Elements of the course. 1. Complements of analysis. 1.1. Ekeland’s variational principle; 1.2. Differentiability; 1.3. Convex functions. 2. Introduction to optimization. 2.1. Optimization with constraints; 2.2. Separation and duality theorems. 3. Introduction to linear programming. 3.1. The problem of linear programming; 3.2. Duality in linear programming; 3.3. Perturbation. 4. Optimality conditions. 4.1. First-order necessary conditions; 4.2 Second-order conditions; 4.3. Lagrangian duality. 5. Introduction to Hilbert spaces. 5.1. Basic definitions; 5.2. The projection theorem; 5.3. Hilbert bases. 6. Introduction to variational formulation of some problems. II. Exercises and problems. 7. Exercises in finite dimension. The 68 exercises posed in the finite-dimension setting deal with geometric objects such as convex sets, cones and their polars, polyhedra, support functions etc., classical minimization problems (the area of a triangle of a given perimeter, the areas of lateral faces of a tetrahedron, Pythagoras’ theorem in 3D, dots), and also with classical results such as Farkas’ and Gordan’s lemmas and Caratheodory’s and Minkowski’s theorems, or applications to problems related with energy, entropy, the regression problem in statistics, the choice of the best financial investment, etc. 8. Exercises in infinite dimension. Some of the 35 proposed problems in this infinite setting are related with: the regularization for convolution, Sobolev’s spaces, problems related with polar cones, the problem of the brachistochrone, the Ekeland variational principle and several applications, the strong and weak convergence in Hilbert spaces, problems about projections, the separation of a convex function, a concave function, etc. (ZDM/Mathdi)