As the foundation of SVMs, the optimization fundamentals are introduced in this chapter. It includes two parts: the basic part — Sections 1.1-1.3 and the advanced part — Sections 1.4-1.5. Sections 1.1, 1.2 and Section 1.3 are respectively concerned with the traditional convex optimization in Euclidian space and Hilbert space. For the readers who are not interested in the strict mathematical argument, Section 1.3 can be read quickly just by comparing the corresponding conclusions in Hilbert space and the ones in Euclidian space, and believing that the similar conclusions in Hilbert space are true. Sections 1.4-1.5 are mainly concerned with the conic programming and can be skipped for those beginners since they are only used in the later subsections 8.4.3 and 8.8.4. In fact they are mainly served for further research. We believe that, for the development of SVMs, many applications of conic programming are still waiting to be discovered.