The concept underlying the support vector machine (SVM) is to find a “plane” that can separate samples belonging to different categories. This chapter will start a tour to SVM with an introduction to the basic SVM. The idea of SVM will be explained using mathematical functions and graphical illustrations. We will see how to derive the optimization formulation of a machine learning problem using SVM. To facilitate solution, the conversion of this original formulation of the optimization problem into a dual problem will be explained. Based on that, we will show how to generalize the basic SVM for linear problems via kernel functions to obtain nonlinear SVMs. After that, soft margin as a common technique for addressing overfitting issues in SVM applications will be introduced. Like the basic SVM (with soft margin), the introduction to SVM with hard margin will also include formulations of the original optimization problem and dual problem. In the final, extra skills about SVM, including the SMO algorithm and the use of SVM for multi-class classification and regression, will be briefly discussed.