The Work presented in this book is divided into two parts. The first part of our work is confined to the study of equilibrium properties of continuous spin systems using different Monte Carlo (MC) techniques. The other part of the work involves the computer simulation of equilibrium and non-equilibrium properties of a number of continuous spin models. In equilibrium study, we have performed both one dimensional and two dimensional random walk in energy space as well as energy-order parameter & energy-correlation function space to generate the density of states and Joint Density of states of a system using the Wang-Landau (WL) algorithm and Wang-Landau-Transition-Matrix (WLTM) algorithm. In the second part of the work, our aim is to determine the local and global persistence exponents associated with zero temperature quenched dynamics of the two-dimensional XY model and the three-dimensional Heisenberg model using the method of block persistence in a single simulation.