Let $G$ be a semisimple linear algebraic group defined over an algebraically closed field $k$ . Fix a smooth projective curve $X$ defined over $k$ , and also fix a closed point $x\in X$ . Given any strongly semistable principal $G$ -bundle $E_G$ over $X$ , we construct an affine algebraic group scheme defined over $k$ , which we call the monodromy of $E_G$ . The monodromy group scheme is a subgroup scheme of the fiber over $x$ of the adjoint bundle $E_G\times^G G$ for $E_G$ . We also construct a reduction of structure group of the principal $G$ -bundle $E_G$ to its monodromy group scheme. The construction of this reduction of structure group involves a choice of a closed point of $E_G$ over $x$ . An application of the monodromy group scheme is given. We prove the existence of strongly stable principal $G$ -bundles with monodromy $G$