General Relativity (GR) is one theory amongst a wider range of plausible descriptions of the Universe. The aim of this thesis is to examine the behaviour of so-called screened theories, which are designed to avoid local tests of modified gravity (MG). We establish that these theories may be treated in a unified manner in the context of halo formation. A prerequisite for this is the clarification that the quasi-static approximation can be applied in cosmologically-plausible scenarios. Amongst the plethora of MG theories, we select three, each of which exhibit a different form of screening. This describes a self-concealing property whereby each theory behaves like GR in the conditions of the local Universe. Only at regions of high energy density (chameleon), large coupling to matter (symmetron) or large derivatives of the scalar field (Vainshtein) does their modified behaviour emerge. We examine f(R), symmetron and DGP gravity in the context of non-linear gravitational collapse for the remainder of the thesis. Relativistic scalar fields are ubiquitous in our modern understanding of structure formation. They arise as candidates for dark energy and are at the heart of many modified theories of gravity. While there has been tremendous progress in calculating their effects on large scales there are still open questions on how to best quantify their effects on smaller scales where non-linear collapse becomes important. In these regimes, it has become the norm to use the quasi-static approximation in which the time evolution of perturbations in the scalar fields are discarded, akin to what is done in the context of non-relativistic fields in cosmology and the corresponding Newtonian limit. We show that considerable care must be taken in this regime by studying linearly perturbed scalar field cosmologies and quantifying the error that arise from taking the quasi-static limit. We focus on f(R) and chameleon models to assess the impact of the quasi-static approximation and discuss how it might affect studying the non-linear growth of structure in N-body numerical simulations. The halo mass function (HMF) n(M) dM is the number of haloes with mass in the range [ M, M+dM ] per unit volume. It has two remarkable properties which render it a useful probe of extensions to general relativity (GR). On the one hand, it is (nearly-)universal, in the sense that it can be written in a form (f(v) which is (practically) insensitive to changes in redshift and cosmological parameters and redshift. We develop a method to generalise fitting functions derived in GR to a variety of screened MG theories, in order to examine whether they are universal in the sense of being insensitive to MG. On the other hand, the HMF is sensitive to both the expansion history of the universe and the non-linear behaviour of spherical collapse via the critical density parameter and the matter power spectrum via the halo resolution. This greatly complicates the theoretical framework required to calculate the HMF, particularly given the sensitivity of chameleon MG to the surrounding environment. We explore a variety of new and existing methods to do so. Finally we re-calibrate the MG halo mass functions with the same rigour as has been done in GR. An important indicator of modified gravity is the effect of the local environment on halo properties. This paper examines the influence of the local tidal structure on the halo mass function, the halo orientation, spin and the concentration-mass relation. We generalise the excursion set formalism to produce a halo mass function conditional on large-scale structure. Our model agrees well with simulations on large scales at which the density field is linear or weakly non-linear. Beyond this, our principal result is that f(R does affect halo abundances, the halo spin parameter and the concentration-mass relationship in an environment-independent way, whereas we find no appreciable deviation from LCDM for the mass function with fixed environment density, nor the alignment of the orientation and spin vectors of the halo to the eigenvectors of the local cosmic web. There is a general trend for greater deviation from LCDM in under-dense environments and for high-mass haloes, as expected from chameleon screening. Given the broad spectrum of MG theories, it is important to design new probes of MG. Despite the fact that we examine only three theories of MG, the techniques and methodology developed in this thesis can be applied to a wide variety of theories and can be extended to improve the results in this work.

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