An algebras is a vector spaces together with a multiplication similar to, and modeled on, the multiplication available to the vector space of square matrices. In this chapter we develop many of their properties and discuss complex numbers, quaternions and octonions from this perspective. We prove the Cayley-Dickson theorem and Hurwitz’s theorem. We discuss the Pauli spin matrices, the Dirac algebra and Clifford algebras in general. The chapter finishes with a look at matrix Lie groups and their algebras and the groups associated with preserving a bilinear or sesquilinear form.