M-theory is a quantum theory of gravity in 11 dimensions that could possibly describe our world. Unfortunately, it is difficult to study and little is known about it beyond a few quantities that are protected by supersymmetry. The AdS/CFT conjecture provides a non-perturbative definition of M-theory, by relating a stack of N M2-branes in M-theory to a family of maximally supersymmetric conformal field theories (SCFTs) in three dimensions with U(N)xU(N) gauge group called ABJM theory. When N is large, ABJM theory is dual to classical supergravity in four dimensions with a negative cosmological constant, where sub-leading terms in N correspond to corrections from M-theory. ABJM theory itself is strongly coupled, and so is difficult to study using conventional methods. In this thesis, we use the recently discovered conformal bootstrap technique to compute quantities in ABJM theory, and thus M-theory via the AdS/CFT correspondence.
We begin by deriving a protected 1d topological sector in all 3d SCFTs with half maximal supersymmetry, which we use to compute certain protected observables in ABJM theory exactly for low N as well as in a large N expansion to all orders in 1/N. For N=3, we find a new duality between ABJM theory and another kind of maximally supersymmetric 3d SCFT that previously had no M-theory interpretation. We then use the conformal bootstrap to compute numerical bounds on observables in all maximally supersymmetric 3d SCFTs, and find that the previous analytic results for ABJM theory come close to saturating these bounds, which allows us to conjecturally read off the low-lying spectrum of this theory for all N. We then use the Mellin space formalism to compute the same quantities on the AdS side, and find that to leading order in N they match the predictions from the conformal bootstrap, providing a new check of the AdS/CFT conjecture. Finally, we outline a strategy to derive the M-theory S-Matrix from ABJM theory to all orders in N, and check that it works to sub-leading order in N using the previously derived analytic results.