Kelvin-Helmholtz instability at the interface between two separate fluids that move in two opposite directions is considered. In the simplest condition, two similar inviscid fluids with a non-rotary flow are considered on both sides. In this project, Kelvin-Helmholtz instability has been studied for the interface between two non-mixing fluids and different densities. Dimensionless parameters in this problem are Weber number and Reynolds number for studying the effect of surface tension and viscosity respectively. Numerical simulations were performed using a finite difference/Front tracking method. The behavior of the interface is studied at different flow conditions. Specifically, the spreading of an applied disturbance wave on the interface is studied. The probability of drop formation is investigated in the flow. In order to analyze the problem and develop a desirable computer code, a rectangular domain with periodic horizontal boundary conditions and rigid moving walls in its upper and lower parts are considered. Numerical method used for the computations presented here is based on writing one set of equations for the entire computational domain. The momentum equation is discretized on a regular staggered grid using second-order central differences for the spatial derivatives and a second-order predictor–corrector time integration scheme. The continuity equation, when combined with the momentum equation results in a pressure equation that is not separable as for homogeneous flow and is solved by a multi-grid method. The interface development between two fluids can be determined by the velocity difference, surface tension, density, and the viscosity of fluids. When the viscosities are adequately small, it is expected that the initial growth rate is predictable by the linear stability analysis of inviscid fluids. Several simulations of the two-dimensional Kelvin– Helmholtz instability of immiscible fluids are done. The Reynolds numbers selected are sufficiently high so that the initial instability is well predicted by inviscid linear stability theory. It can be easily noticed that when we consider large Weber numbers, the initial disturbance grows rapidly. It has been observed that as the Weber number increases, the wave grows more rapidly, and the fluids grow into each other in a symmetric way for unity density ratio due to low surface tension. For density ratios more than one, we can see asymmetric penetration in the flows. If the lower flow retains a higher Reynolds number, due to the reduction of its viscosity, it is noticed that the lower fluid penetrates into the upper fluid, and there is an asymmetry in the flow. In this research, it is attempted to find the condition where the interface would create discrete zones (drops). In all conditions that was explored here, this phenomenon is not observed for Weber numbers lower than 4. In other words, drop formation does not occur for Weber numbers below 4. For Weber numbers around 6 (5 to 7), the interface is capable to form drops, and release it into the other fluid. At larger Weber numbers (larger than 7), the growth rate of the interface is like fingers. These fingers or filaments form in the heavier fluid and release in the lighter fluid. Ultimately, the effects of surface tension and viscosity on the development of the interface have been discussed. Key Words: Kelvin-Helmholtz Instability, Two-Phase Flow, Shear Flow, Surface Tension, Drop.