The nature of gravity as we know is that it pulls. Issac Newton’s Principia generalised this gravity as a universal attractive force which weakens with distance. However in the late 1990s it was found that our Universe is undergoing an accelerated expansion, which has since then been confirmed by numerous independent cosmological probes. Thus on the scale of the Universe, we need a repulsive force. This acceleration is an unsolved puzzle which is one of the most exciting topics in cosmology. The accelerated expansion could be due to an unknown exotic component, which has been christened as the ‘dark energy’, or it could be that on the large scales there is a breakdown of our understanding of the gravitational theory. One of the observational methods through which we can test the theory of gravity at those scales is known as ‘redshift-space distortions’ (RSD), which is the main topic of this thesis. To probe the three-dimensional distribution of galaxies in the Universe, we require the redshift as the radial distance indicator in addition to the two-dimensional angular position in the sky. The recession velocity of any galaxy is proportional to the distance from us according to the Hubble expansion law. However the observed velocity has an additional contribution from peculiar velocities. These are generated due to the clustering dynamics and thus contaminate the distance information. Thus RSD causes a change in the clustering pattern as compared to the actual galaxy distribution. The main aim of this thesis is to model the n-point spatial clustering statistics in redshift-space faithfully. The main motivation factor in undertaking this task is that the future redshift surveys will offer us an influx of (big) data as never seen before. For example LSST is poised to produce about 20 terabytes of raw data per night and about 60 petabytes of data during its operation period. However to take advantage of this plethora of data, we need accurate theoretical models. In chapter 2, we focus on the two-point clustering information. The state-of-the-art constraints on the gravitational theory based on RSD come from the so called ‘Gaussian streaming model’ (GSM). We find that the success of the GSM appears to be fortuitous. We improve upon this by introducing the ‘generalised hyperbolic streaming model’, which we showcase to work extremely well at non-linear scales and does not have to rely on any fortuitous cancellations. The future redshift surveys will be able to measure higher-order correlation functions precisely. In light of this we develop the exact n-point streaming model in chapter 3. We also introduce a phenomenological model based on the GSM tailored for three-point clustering information in redshift space, in which all the ingredients were taken from the linear perturbation theory. This paves way for extracting the cosmological information from higher-order clustering statistics in a precise manner. We also look at the effect of RSD on pairwise velocity statistics measured from the kinetic Sunyaev-Zeldovich effect in chapter 4, which will be measured precisely in future cosmic microwave background (CMB) experiments. We see that the mean pairwise velocity in redshift space undergoes a sign inversion and this comes about from the ‘Finger-of-God’ (FoG) effect. We also explore the effects of RSD on the three-point velocity statistics and further construct an estimator which is ideal to extract the cosmological parameters.