<p> This dissertation presents a new high-order front tracking method for two-phase hyperbolic systems of conservation laws separated by a contact discontinuity. A review of existing methods for moving and/or irregular boundaries shows the significance of accurate geometry data and flux calculation near the interface to achieve a high order method. A general method for hyperbolic systems of conservation laws is presented along with the implementations of numerical methods for simulations of gas dynamics in 2-D using the Euler equations. Convergence tests show the new method is second order accurate for smooth solutions and first order in presence of shocks. Also the new method is used for simulation of Richtmyer-Meshkov instability, in which results are in agreement with both theoretical and experimental approaches.</p>