The study of dynamics of multiphase and multicomponent flows is one of great importance in fluid mechanics duo to the economic significances and wide engineering applications of such flows in petroleum industry, simulation of sprays, droplets and bubbles dynamics, to name a few. Duo to this wide spectrum of scientific and industrial applications, the numerical simulation of multiphase and multicomponent fluid flows is critically important. The dynamics of such flows involves several complexities duo to the complex interaction of several physical phenomena such as surface tension, phase transition and so on, which all should be accounted for in a simulation. A quick review on the literature shows that the lattice Boltzmann method (LBM) has been successfully employed to simulate a variety of complex fluid flows such as multiphase flows in porous media. For such simulations, several lattice Boltzmann models based on the color gradient, the free-energy (FE) and the Shan-Chen (SC) model have been developed. In this thesis, we first present a brief review of the LBM. Then the algorithms of the original SC and the FE model of Zheng et al. models are presented in details. These models are then studied numerically to evaluate their performance in the modeling of several 2D immiscible two phase flows. To validate our code for the FE model, the Laplace and the square droplet tests are performed. Unlike some other existing FE models, it is shown that the present model is able to satisfy the Galilean invariance. The deformation and breakup of a droplet in a simple shear flow is also simulated by the FE model. Our results show that this model is able to simulate this problem in a wide range of the Reynolds and Capillary numbers. To investigate the performance of the SC model, the square droplet test is done. It is not possible to specify the surface tension in the original SC model, and hence, one needs to do the Laplace test to measure this property for different problems. It appears that the original SC model suffers from some limitations particularly in specifying proper values for some thermo physical properties such as the surface tension and the viscosity. Duo to this shortcoming of the SC model, it is shown that for the simulation of a droplet in a simple shear flow, it may not be possible to run the simulation for a wide range of the Reynolds and Capillary numbers. To compare the performance of the SC and the FE models, different benchmarks have also been studied. It is shown that both models produce comparable results for these benchmarks. In terms of accuracy and the stability, the FE model seems to perform better compared to the SC model. It is also observed that the FE model is able to simulate multicomponent flows with higher density ratios compared to the SC model. Keywords: Lattice Boltzmann method, Shan-Chen model, Free-energy model, Droplet deformation in simple shear flow