  Let $X$ be a geometrically irreducible smooth projective curve defined over a
field $k$. Assume that $X$ has a $k$-rational point; fix a $k$-rational point
$x\in X$. From these data we construct an affine group scheme ${\mathcal G}_X$
defined over the field $k$ as well as a principal ${\mathcal G}_X$-bundle
$E_{{\mathcal G}_X}$ over the curve $X$. The group scheme ${\mathcal G}_X$ is
given by a ${\mathbb Q}$--graded neutral Tannakian category built out of all
strongly semistable vector bundles over $X$. The principal bundle $E_{{\mathcal
G}_X}$ is tautological. Let $G$ be a linear algebraic group, defined over $k$,
that does not admit any nontrivial character which is trivial on the connected
component, containing the identity element, of the reduced center of $G$. Let
$E_G$ be a strongly semistable principal $G$-bundle over $X$. We associate to
$E_G$ a group scheme $M$ defined over $k$, which we call the monodromy group
scheme of $E_G$, and a principal $M$-bundle $E_M$ over $X$, which we call the
monodromy bundle of $E_G$. The group scheme $M$ is canonically a quotient of
${\mathcal G}_X$, and $E_M$ is the extension of structure group of
$E_{{\mathcal G}_X}$. The group scheme $M$ is also canonically embedded in the
fiber ${\rm Ad}(E_G)_{x}$ over $x$ of the adjoint bundle.
