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Planar maximally supersymmetric Yang-Mills theory (N=4 SYM) is a specialquantum field theory. A few of its remarkable features are conformal symmetryat the quantum level, evidence of integrability and, moreover, it is a primeexample of the AdS/CFT duality. Triggered by Witten's twistor string theory,the past 15 years have witnessed enormous progress in reformulating this theoryto make as many of these special features manifest, from the choice ofconvenient variables to recursion relations that allowed new mathematicalstructures to appear, like the Grassmannian. These methods are collectivelyreferred to as on-shell methods. The ultimate hope is that, by understandingN=4 SYM in depth, one can learn about other, more realistic quantum fieldtheories. The overarching theme of this thesis is the investigation of howon-shell methods can aid the computation of quantities other than scatteringamplitudes. In this spirit we study form factors and correlation functions,said to be partially and completely off-shell quantities, respectively. Moreexplicitly, we compute form factors of half-BPS operators up to two loops, andstudy the dilatation operator in the SO(6) and SU(2|3) sectors using techniquesoriginally designed for amplitudes. A second part of the work is dedicated tothe study of scattering amplitudes beyond the planar limit, an area of researchwhich is still in its infancy, and not much is known about which specialfeatures of the planar theory survive in the non-planar regime. In thiscontext, we generalise some aspects of the on-shell diagram formulation ofArkani-Hamed et al. to take into account non-planar corrections.

On-shell methods for off-shell quantities in N=4 Super Yang-Mills: from scattering amplitudes to form factors and the dilatation operator