This chapter is devoted to vector algebra and vector calculus. In Sect. 1.1, we give an overview on vector algebra, including definition, Cartesian bases and vector expansion, vector arithmetic operators, and position and distance vectors. Then, Sect. 1.2 discusses the various coordinates systems encountered in this book (e.g., Cartesian, cylindrical, and spherical), as well as transformation among them. We then discuss vector calculus in Sect. 1.3, including vector integral calculus, and vector differential calculus. These are discussed first in Cartesian, cylindrical, and spherical coordinate systems, and then are extended to a general curvilinear coordinate system in Sect. 1.4. Finally, time-harmonic vectors is discussed in Sect. 1.5. Useful computer programs are given in the appendix at the end of the chapter.