Multi component fluid instability problems suffer artificial mass diffusion when numerical methods do not correct for numerical dissipation at an interface. Front tracking provides a sharp interface between components but can be computationally intensive. Adaptive mesh refinement (AMR) is a method of increasing resolution only where needed, which can be more computationally efficient. Under certain conditions AMR can achieve a high resolution numerical solution that would be otherwise unattainable on a uniform grid. In this thesis, we combine front tracking and AMR and apply the new algorithm to fluid mixing problems. A series of timed simulations show that this algorithm can be faster then uniform grid front tracking. The front tracked interface is less diffuse than an AMR calculation at equivalent resolution without front tracking. These results show that combining AMR and front tracking is feasible and the strengths of both methods are retained. The combined algorithm assumes front information resides only at the finest level, simplifying the code and easing the way for future modifications.