In these notes we describe the classical groups, that is, the linear groups and the orthogonal, symplectic, and unitary groups, acting on finite dimensional vector spaces over skew fields, as well as their pseudo-quadratic generalizations. Each such group corresponds in a natural way to a point-line geometry, and to a spherical building. The geometries in question are projective spaces and polar spaces. We emphasize in particular the rôle played by root elations and the groups generated by these elations. The root elations reflect — via their commutator relations — algebraic properties of the underlying vector space.

We also discuss some related algebraic topics: the classical groups as permutation groups and the associated simple groups. I have included some remarks on K-theory, which might be interesting for applications. The first K-group measures the difference between the classical group and its subgroup generated by the root elations. The second K-group is a kind of fundamental group of the group generated by the root elations and is related to central extensions. I also included some material on Moufang sets, since this is an interesting topic. In this context, the projective line over a skew field is treated in some detail, and possibly with some new results. The theory of unitary groups is developed along the lines of Hahn & O'Meara. Other important sources are the books by Taylor and Tits, and the classical books by Artin and Dieudonné.