We define the stack of G-bundles over smooth projective curves for a reductive algebraic group G as well as the stack of quasi-parabolic G-bundles over s-pointed smooth curves. Both of these stacks are smooth algebraic stacks. We define a tautological family of G-bundles over any smooth curve ? parameterized by the infinite Grassmannian. Then, we prove the Uniformization Theorem for both of the above two stacks. An important ingredient in the proof of the above two uniformization theorems is a result due to Drinfeld--Simpson asserting that for a family of G-bundles over ? parameterized by a scheme S, the pull-back of the family to some étale cover is trivial restricted to any affine open subset of ?. In particular, the uniformization theorems give a bijective parameterization of G-bundles as well as quasi-parabolic G-bundles over ?.