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I have some manifold $M$ and am wondering what kind of Principal Bundles I am allowed to construct on it.

To be more precise, what are the restrictions when trying to construct principal Bundles over some Manifold? I imagine the topological properties give some quite strict restrictions, but I couldn't find anything in the literature I own.

I am specifically looking for restrictions found on the Torus $T^2$. Any pointers are greatly appreciated!

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    This might need some fleshing out, but I think $\pi_1X$ (and maybe $\pi_1G$) might be the only obstruction to constructing principal bundles over a surface, if you construct the bundle step by step over a skeleton of the CW structure.2012-05-08

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Isomorphism classes of principal $G$-bundles over $T^2$ are labeled by $\pi_1(G)$. Please, see the following article by: Klimek-Chudy and Kondracki: "The topology of the Yang-Mills theory over torus", where the isomorphism classes are computed for a few low dimensional base manifolds.

For an alternative method of computation, please see the following two articles by: Yu. A. Kubyshin: arXiv:math/9911217: "A classification of fibre bundles over 2-dimensional spaces", and arXiv:math-ph/0309059: "Geometrical formalism in gauge theories".