  When the action of a reductive group on a projective variety has a suitable
linearisation, Mumford's geometric invariant theory (GIT) can be used to
construct and study an associated quotient variety. In this article we describe
how Mumford's GIT can be extended effectively to suitable actions of linear
algebraic groups which are not necessarily reductive, with the extra data of a
graded linearisation for the action. Any linearisation in the traditional sense
for a reductive group action induces a graded linearisation in a natural way.
The classical examples of moduli spaces which can be constructed using
Mumford's GIT are moduli spaces of stable curves and of (semi)stable bundles
over a fixed nonsingular curve. This more general construction can be used to
construct moduli spaces of unstable objects, such as unstable curves or
unstable bundles (with suitable fixed discrete invariants in each case, related
to their singularities or Harder--Narasimhan type).
