  In this article we review the question of constructing geometric quotients of
actions of linear algebraic groups on irreducible varieties over algebraically
closed fields of characteristic zero, in the spirit of Mumford's geometric
invariant theory (GIT). The article surveys some recent work on geometric
invariant theory and quotients of varieties by linear algebraic group actions,
as well as background material on linear algebraic groups, Mumford's GIT and
some of the challenges that the non-reductive setting presents. The earlier
work of two of the authors in the setting of unipotent group actions is
extended to deal with actions of any linear algebraic group. Given the data of
a linearisation for an action of a linear algebraic group H on an irreducible
variety $X$, an open subset of stable points $X^s$ is defined which admits a
geometric quotient variety $X^s/H$. We construct projective completions of the
quotient $X^s/H$ by considering a suitable extension of the group action to an
action of a reductive group on a reductive envelope and using Mumford's GIT. In
good cases one can also compute the stable locus $X^s$ in terms of stability
(in the sense of Mumford for reductive groups) for the reductive envelope.
