  In this Master of Science Thesis I introduce geometric algebra both from the
traditional geometric setting of vector spaces, and also from a more
combinatorial view which simplifies common relations and operations. This view
enables us to define Clifford algebras with scalars in arbitrary rings and
provides new suggestions for an infinite-dimensional approach.
  Furthermore, I give a quick review of classic results regarding geometric
algebras, such as their classification in terms of matrix algebras, the
connection to orthogonal and Spin groups, and their representation theory. A
number of lower-dimensional examples are worked out in a systematic way using
so called norm functions, while general applications of representation theory
include normed division algebras and vector fields on spheres.
  I also consider examples in relativistic physics, where reformulations in
terms of geometric algebra give rise to both computational and conceptual
simplifications.
