One of the most exciting and successful ideas pursued in string theory is gauge/gravity duality. We consider the example of the AdS/CFT correspondence, which maps maximally supersymmetric Yang-Mills (N = 4 SYM) in four dimensions with gauge group U(N) to closed strings propagating in a background of Anti de Sitter space crossed with a sphere (AdS5 × S5). Much progress has been made understanding this duality in the planar ’t Hooft limit, where we fix the coupling of the gauge theory λ and take N large. On the gravity side the string coupling gs is proportional to 1/N for fixed λ, so in this limit we get classical string theory. In this thesis we use symmetric group methods to study the AdS/CFT correspondence exactly at finite N, without taking the planar limit. This takes the string theory into the quantum regime and allows us to probe phenomena which are non-perturbative in gs. First we enumerate the spectrum. While the spectrum is non-trivial in the planar limit, it is further complicated at finite N by the Stringy Exclusion Principle, which truncates the usual trace spectrum. We organise local operators in the gauge theory using representations of the gauge group U(N), which for heavy operators are interpreted in terms of giant graviton branes in the bulk. To do this we sort the different fields of the theory into representations of the global superconformal symmetry group using Schur- Weyl duality. We then compute two- and three-point functions of these operators exactly to all orders in N for the free theory and at one loop. We use these correlation functions to resolve certain transition probabilities for giant gravitons using CFT factorisation