Abstarct: Investigation of multi-phase systems such as drop and bubble is one of the research topics of interest. This behavior of these systems is dependent on tensions between the two fluids, inertial force, surface tension, gravity and pressure gradient between the two fluids and is used in the fields of mixing, fermentation, separation, transfer of materials in the chemical industry, oil and gas extraction, cooling, color and print by jet. So far, several research using numerical, analytical and experimental methods in relation to the behavior of multi-phase systems consisting of various fluids are done. Literature review shown that, the investigation of the drop formation and deformation with regard to non-Newtonian behavior of inherent non-Newtonian fluids like oil and emulsion is more practical and reasonable. Lattice Boltzmann method is interface of two major trends of continuum (Macroscopic scale), and discrete (Microscopic scale) in simulating which the behavior of a set of particles can be used. Lattice Boltzmann simulations are validated by using the contact angle test, the flow in the channel as well as two-phase flow analysis in the T-junction for non-Newtonian shear thinning fluid baxsed on Carreau-Yasuda model. Results have a good agreement with the analytical and available results. Newtonian and non-Newtonian dispersed phase and Newtonian and non-Newtonian continuous phase are simulated in T-junction micro-channel. The influence of various parameters including the capillary number, power of Carreau-Yasuda model, the flow rate ratio and viscosity ratio on drop formation was investigated. Results shown that the properties of the continuous phase have a great influence on the formation and droplet size. Newtonian and non-Newtonian single phase fluid and deformation of Newtonian and non-Newtonian drop in Newtonian and non-Newtonian fluid under simple shear flow between two parallel plates that are moving at opposite speeds, have been simulated. Then, the influence of various parameters including the capillary number, Reynolds number, power of Carreau-Yasuda model, viscosity ratio on drop deformation was investigated. By reducing the the power of Carreau-Yasuda model, the elongation of non-Newtonian drop in Newtonian fluid increases and deformation of Newtonian drop in non-Newtonian fluid decreases. Good agreement with literature data and results is observed. The present study can be considered as a development towards a numerical tool to investigate the behavior of Newtonian and non-Newtonian drops in Newtonian and non-Newtonian fluid in shear flow and as a verification and validation step for further applications in more complex flows.