This unique thesis covers all aspects oftheories of gravity beyond Einstein's General Relativity, from setting up theequations that describe the evolution of perturbations, to determining thebest-fitting parameters using constraints like the microwave backgroundradiation, and ultimately to the later stages of structure formation usingstate-of-the-art N-body simulations and comparing them to observations ofgalaxies, clusters and other large-scale structures. This truly ground-breakingwork puts the study of modified gravity models on the same footing as thestandard model of cosmology. Since the discovery of the accelerating expansionof the Universe, marked by the awarding of the 2011 Nobel Prize in Physicsthere has been a growing interest in understanding what drives thatacceleration. One possible explanation lies in theories of gravity beyondEinstein's General Relativity. This thesis addresses all aspects of the probleman approach that is crucial to avoiding potentially catastrophic biases in theinterpretation of upcoming observational missions.

Introduction.- Linear Perturbations in Galileon Gravity Models.- The Observational Status of Galileon Gravity After Planck.- Spherical Collapse in Galileon Gravity .- N-body Simulations and Halo Modelling in Galileon Gravity Cosmologies.- Nonlinear Structure Formation in Nonlocal Gravity.- Lensing by Clusters and Voids in Modi¿ed Lensing Potentials.- Summary, Conclusions and Future Work.