This chapter introduces vectors as mathematical objects characterized by both magnitude and direction, distinguishing them from scalars. Concepts include vector notation, components and operations such as addition, scalar multiplication, dot and cross products and triple products, which are pivotal in physics and engineering. Graphical interpretation using arrows allows an understanding of these operations. Advanced topics cover transformations of vectors under coordinate changes, using tools like transformation matrices, Kronecker delta and Levi–Civita symbols. The discussion extends to the differential calculus of vectors, exploring gradient, divergence, curl and integral formulations (line, surface and volume integrals) critical for analysing physical fields. Special functions like Legendre polynomials and Bessel functions are introduced for problems involving spherical and cylindrical symmetries. The chapter concludes with the applications of vector concepts and specialized functions in electrodynamics, offering a foundation for analysing spatial and directional phenomena.