Support vector machine (SVM) has been attracting considerable attention in the field of pattern recognition, machine learning, and so on. The purpose of this report is to review fundamental principles of the SVM and the decomposition method which is widely used for the training of SVMs. In the first part, some key ideas characterizing SVMs such as margin. maximization, soft margin, and kernel trick are explained. In the second part, the basic idea behind the decomposition method, some representative algorithms, and recent results on its global convergence are introduced.