This thesis is divided into three sections. In the first section, we discuss Singular Inflation, in which a scalar field exhibits a weak singularity, originally illustrated by the non-integer powerlaw potentials. We expand the class of potentials that result in this phenomenon, by providing constraints on the derivatives of the functions. In the second section, we study the cosmological effects of adding terms of the form f(TμνTμν) = η(TμνTμν)n to the matter Lagrangian of general relativity. The resulting cosmological theories give rise to field equations of similar form to several particular theories with different fundamental bases, including bulk viscous cosmology, loop quantum gravity, k-essence, and brane-world cosmologies. We find a range of exact solutions for isotropic universes, discuss their behaviours with reference to the early- and late-time evolution, accelerated expansion, and the occurrence or avoidance of singularities. We briefly discuss extensions to anisotropic cosmologies and delineate the situations where the higher-order matter terms will dominate over anisotropies on approach to cosmological singularities. Finally, in the third section, we study a related model, called Energy-Momentum Log Gravity (EMLG), constructed by the addition of the term f(TμνTμν) = α ln(λ TμνTμν) to the Einstein-Hilbert action with cosmological constant Λ. This choice of model results in constant effective inertial mass density and has an explicit exact solution of the matter energy density in terms of redshift. We look for viable cosmologies, in particular, an extension of the standard ΛCDM model. EMLG provides an effective dynamical dark energy passing below zero at large redshifts, accommodating a mechanism for screening Λ in this region, in line with suggestions for alleviating some of the tensions that arise between observational data sets within the standard ΛCDM model. We present a detailed theoretical investigation of the model and then constrain the free parameter α′, a normalisation of α, using the latest observational data. The data do not rule out the ΛCDM limit of our model (α′ = 0), but prefers slightly negative values of the EMLG model parameter (α′ = −0.032 ± 0.043), which leads to the screening of Λ. We also discuss how EMLG relaxes the persistent tension that appears in the measurements of H0 within the standard ΛCDM model.