The mathematical description of a practical problem as a convex optimization problem is very desirable since for such a program local minimizers and points which satisfy first order necessary optimality conditions are also global minimizers and special numerical methods may be available for its solution. The goal of the book is to thoroughly discuss six relevant classes of problems from the applications which lead to convex optimization problems. These classes comprise the building and training of support vector machines, parameter estimations in connection with maximum likelihood estimation and expectation maximization algorithms, linear minimum norm approximations and regularization techniques, semidefinite programming and linear matrix inequalities with applications in linear control theory, convex relaxations of non-convex problems, and problems from geometry. The author concentrates on the detailed derivation of the formulation of the convex programs, and he thereby intends to also develop the skills of the reader in recognizing convexity in optimization. Some important concepts of convex optimization are briefly summarized in an initial section. It includes a short subsection on the CVX toolbox which allows the author to give simple Matlab descriptions of convex programs.