  The purpose of this work is to describe the (category of) Higgs bundles on a
complex scheme X having a given cameral cover X~. We show that this category is
a T_{X~}-gerbe, where T_{X~} is a certain sheaf of abelian groups on X, and we
describe the class of this gerbe precisely. In particular, it follows that the
set of isomorphism classes of Higgs bundles with a fixed cameral cover X~ is a
torsor over the group H^1(X, T_{X~}), which itself parametrizes T_{X~}-torsors
on X. This underlying group can be described as a generalized Prym variety,
whose connected component is either an abelian variety or a degeneration
thereof.
