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We have performed extensive computer simulations in order to explore the equilibrium and non-equilibrium statistical mechanics of planar (spatial dimension=2) lattice models representing nematic liquid crystal. The first part of our work deals with equilibrium Monte Carlo computer simulation study of the role of topological defects in the equilibrium Nematic to Isotropic phase transition associated with the Lebwohl- Lasher lattice model. We have studied the phase transition in details and evaluated the thermodynamic limit of transition temperature as well as the associated critical exponents using Finite Size Scaling analysis. The second part of our work deals with the studies related to the ordering dynamics of quenched planar uniaxial nematics represented by the Lebwohl-Lasher lattice model. First passage problem in phase ordering kinetics has also been studied. The entire work was submitted as thesis for obtaining Ph.D. degree of Dr. Subhrajit Dutta from Jadavpur University, Kolkata.