We combine supersymmetric localization with the numerical conformal bootstrap to non-perturbatively study 4d N=4 super-Yang-Mills (SYM) and 3d ABJM theory for all N and coupling, which is dual to string theory and M-theory, respectively. For N=4 SYM, our bound on the lowest dimension operator interpolates between weak coupling results for the Konishi operator, and strong coupling results for the lowest double trace operator, including the first few stringy corrections. For ABJM, our bounds match the protected strong coupling results from M-theory, and give the first prediction for the first unprotected correction D^8R^4. In both cases, our results suggest that bootstrap + localization is sufficient to numerically solve holographic theories non-perturbatively, opening a new window on strongly coupled quantum gravity.
