The quest for obtaining higher Tc superconductivity led to the discovery of cuprates about 20 years ago. Since then, they continue to puzzle the scientific community with their bizarre properties like non-BCS superconductivity, pseudo gap, Fermi arcs, linear T resistivity etc. Since these materials show unusually high Tc, a novel mechanism is at play and strong correlations are believed to play an important role. The theme of this thesis work is to study physics of such strongly correlated systems in two dimensions at T = 0 along with development of new theoretical tools necessary for the study. The focus of the thesis is on the ground state studies of strongly correlated models like t-J and Hubbard models using variational Monte Carlo (VMC) and renormalized mean field theory (RMFT). The general method is to propose a variational wave function, motivated by the physics ideas, to be a candidate ground state of the system. Methods to efficiently evaluate the ground state energy and minimizing it with respect to the variational parameters are developed in this work. Antiferromagnetism-superconductivity competition and electron-hole asymmetry in the extended t-J model is investigated. The main result of this work is that increasing the magnitude of the next neighbor hopping (t') on hole doped side strengthen superconductivity while it stabilizes antiferromagnetism on the electron doped side. It is also shown that it is possible to characterize the T = 0 phase diagram with just one parameter called as Fermi Surface Convexity Parameter (FSCP). Next, the possibility of phase separation in the t-J model on a square lattice is investigated using local RMFT technique. It is found that for certain doping, the system phase separates into regions with antiferromagnetic and superconducting orders. Next, the role played by crystalline anisotropy of orthorhombic YBCO cuprates on their properties is examined using anisotropic tx-ty-J model and this ground state study suggests that the anisotropies seen in their properties are plausible solely due to the crystalline anisotropy. A new general method to study strongly correlated systems with singlet ground states is developed and tested in this thesis work. The last part of the thesis explores the possibility of high Tc superconductivity in graphene which is a intermediate coupling resonating valence bond (RVB) system. It is found that undoped graphene is not a superconductor, consistent with the experiments. On doping, the ground state of graphene is found to be a superconductor with “d+id” symmetry whose strength shows a dome as a function of doping which is reminiscent of RVB physics.