After recalling the basic notions concerning pro nite and
proalgebraic group completions and Tannakian categories, we review
how the latter can be used to de ne generalizations of the notion of
fundamental group of a space, such as the Nori and Langer fundamental
groups, and the algebraic fundamental group introduced by Simpson.
Then we discuss how one can de ne a Tannakian category whose objects
are Higgs bundles on a complex projective variety that are \numerically
at" in a suitable sense, and show how the Higgs fundamental group is
related to a conjecture about semistable Higgs bundles.
