Cosmology has become a precision science due to a wealth of new precise data from various astronomical observations. It is therefore important, from a methodological point of view, to develop new statistical and numerical tools to study the Cosmic Microwave Background (CMB) radiation and Large Scale Structure(LSS), in order to test different models of the Universe. This is the main aim of this thesis. The standard inflationary $\Lambda$-dominated Cold Dark Matter ($\Lambda$CDM) model is based on the premise that the Universe is statistically isotropic and homogeneous. This premise needs to be rigorously tested observationally. We study the angular correlation function $C(\theta )$ of the CMB sky using the WMAP 5-year data, and find that the low-multipoles can be reconstructed from the data outside the sky cut. We apply a Bayesian analysis and find that $S1/2$ statistic ($S1/2=\int [C(\theta )]2d\cos \theta$, used by various investigators as a measure of correlations at large angular scales) cannot exclude the predictions of the $\Lambda$CDM model. We clarify some issues concerning estimation of correlations on large angular scales and their interpretation.

To test for deviation from statistical isotropy, we develop a quadratic maximum likelihood estimator which we apply to simulated Planck maps. We show that the temperature maps from Planck mission should be able to constrain the amplitude of any spherical multipole of a scale-invariant quadrupole asymmetry at the $1\%$ level ($2\sigma$). In addition, polarization maps are also precise enough to provide complimentary constraints. We also develop a method to search for the direction of asymmetry, if any, in Planck maps.

B-mode polarisation of the CMB provides another important test of models of the early Universe. Different classes of models, such as single-field inflation, loop quantum cosmology and cosmic strings give speculative but testable predictions. We find that the current ground-based experiments such as BICEP, already provided fairly tight constraints on these models. We investigate how these constraints might be improved with future observations (e.g. Planck, Spider).

In addition to the CMB related research, this thesis investigates how peculiar velocity fields can be used to constrain theoretical models of LSS. It has been argued that there are large bulk flows on scales of $\gtrsim 50$ Mpc/h. If true, these results are in tension with the predictions of the $\Lambda$CDM model. We investigate a possible explanation for this result: the unsubtracted intrinsic dipole on the CMB sky may source this apparent flow, leading to the illusion of the tilted Universe. Under the assumption of a superhorizon isocurvature fluctuation, the constraints on the tilted velocity require that inflation lasts at least 6 e-folds longer (at the 95% confidence interval) than that required to solve the horizon problem.

Finally, we investigate Cosmic Mach Number (CMN), which quantifies the ratio between the mean velocity and the velocity dispersion of galaxies. We find that CMN is highly sensitive to the growth of structure on scales $(10,150)$ Mpc/h, and can therefore be used to test modified gravity models and neutrino masses. With future CMN data, it should be possible to constrain the growth factor of linear perturbation, as well as the sum of the neutrino mass to high accuracy.