Much of the recent progress in cosmology has come from studying the power spectrum of the cosmic microwave background (CMB). The latest results from the Planck satellite confirmed that the inflationary paradigm with the $\Lambda$CDM six-parameter model provides a very good description of the observed structures in the Universe. Even so, additional parameters, such as cosmic defects, are still allowed by current observational data. Additionally, many of the inflationary models predict a significant departure from Gaussianity in the distribution of primordial perturbations. Higher order statistics, such as the bispectrum, are required to test and constrain such models. The late-time distribution of matter in the Universe - large-scale structure (LSS) - contains much more information than the CMB that has not yet been used. In this thesis, we look at both problems: the effects of cosmic defects, in particular cosmic strings and domain walls on the CMB power spectrum through numerical simulations, and the dark matter bispectrum of large-scale structure. Topological defects are predicted by most inflationary theories involving symmetry breaking in the early Universe. In this thesis we study the effects of cosmic strings and domain walls on the CMB by determining their power spectrum. We use Nambu-Goto and field theory simulations for cosmic strings and domain walls respectively, and we determine the power spectra they produce with a modified Einstein-Boltzmann solver sourced by unequal time correlators from components of the energy-momentum tensor of the defects. We use these spectra together with CMB likelihoods to obtain constraints on the energy scales of formation of the cosmic defects, finding and MeV (at 95% confidence level) for cosmic strings and domain walls respectively, when using the Planck satellite likelihoods. For the matter bispectrum of LSS, we compare different perturbative and phenomenological models with measurements from $N$-body simulations by using shape and amplitude correlators and we determine on which scales and for which redshifts they are accurate. We propose a phenomenological ‘three-shape’ model, based on the fundamental shapes we have observed by studying the halo model that are also present in the simulations. When calibrated on the simulations, this model accurately describes the bispectrum on all scales and redshifts considered, providing a prototype bispectrum HALOFIT-like methodology that could be used to describe and test parameter dependencies.