  An almost Abelian Lie group is a non-Abelian Lie group with a codimension 1
Abelian normal subgroup. The majority of 3-dimensional real Lie groups are
almost Abelian, and they appear in all parts of physics that deal with
anisotropic media - cosmology, crystallography etc. In theoretical physics and
differential geometry, almost Abelian Lie groups and their homogeneous spaces
provide some of the simplest solvmanifolds on which a variety of geometric
structures such as symplectic, K\"ahler, spin etc., are currently studied in
explicit terms. Recently, almost Abelian Lie algebras were classified and
studied in details. However, a systematic investigation of almost Abelian Lie
groups has not been carried out yet, and the present paper is devoted to an
explicit description of properties of this wide and diverse class of groups.
  The subject of investigation are real almost Abelian Lie groups with their
Lie group theoretical aspects, such as the exponential map, faithful matrix
representations, discrete and connected subgroups, quotients and automorphisms.
The emphasis is put on explicit description of all technical details.
