The first half of chapter 1 exposes the most significant grey areas in the Standard Hot Big Bang model: (a) determining the form and density of the energy content of the Universe, and reconciling this to (b) the age of the Universe, and (c) the observed clustering of galaxies. This thesis makes two contributions to this area. Chapter 2 discusses a variant of the Cold Dark Matter model in which a dark matter component decays radiatively at early times. The model has the virtue that it can accommodate the low value of Ωh inhered from observed large-scale galaxy clustering and the high measured values of Ω and h. Limits on the small-scale clustering predicted by such models constrain the mass and lifetime of the decaying component to 0.5 < m M 30 keV, 0.2 < τ < 500 years. Chapter 3 contributes to the observational tests of large-scale galaxy clustering by constraining the clustering signal of a sample of high-redshift radio galaxies. The high redshift data are important as they give clues to the evolution of the density field with time. Recent work with comparable datasets has measured clustering of a surprisingly high amplitude for such redshifts - such measurements are found to be incompatible with our data. The second half of chapter 1 describes promising theories which extend the Standard Model - inflationary and topological defect models. Although inflation solves some important problems the model is poorly motivated in terms of currently understood particle physics. Furthermore, conclusive tests of inflation are elusive. Topological defects, while less of a panacea for the problems of the field, are well motivated by theories of Grand Unification at T ~ 10¹⁶ GeV and make plausible candidates for the source of primordial inhomogeneities. Crucially, cosmic strings, the best investigated class of topological defect models, have testable consequences for the microwave background and the lensing of galaxies.