  The purpose of this paper is to introduce the notion of mixed twistor
structure, a generalization of the notion of mixed Hodge structure. The utility
of this notion is to make possible a theory of weights for various things
surrounding arbitrary representations of the fundamental group of a smooth
projective variety. We give some examples of generalizations of classical
results for variations of mixed Hodge structure, to the twistor setting. This
supports a ``meta-theorem'' (which we state but don't prove) that one can
everywhere replace the word ``Hodge'' by the word ``twistor''.
  We show that the jet spaces of hyperk\"ahler or more generally hypercomplex
manifolds have natural mixed twistor structures which determine the
hypercomplex structure in a formal neighborhood of a point.
