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Since the mid 1990’s, the leading candidate for a unified theory of all fundamental physical interactions has been M Theory.

A full formulation of M Theory is still not available, and it is only understood through its limits in certain regimes, which are either one of five 10-dimensional string theories, or 11-dimensional supergravity. The equations for these theories are mathematically interesting in themselves, as they reflect, either directly or indirectly, the presence of supersymmetry. We discuss recent progresses and open problems about two of these theories, namely supersymmetric compactifications of the heterotic string and of 11-dimensional supergravity. This is based on joint work of the speaker with Sebastien Picard and Xiangwen Zhang, and with Teng Fei and Bin Guo.