Fluid instabilities, particularly interfacial instabilities, have proven to be a powerful mechanism in driving and sustaining combustion processes in several devices of practical interest. Modern combustors are in fact designed to exploit the mixing and combustion characteristics associated with a broad class of canonical, interfacial instabilities. In spite of their relevance to combustor design, a detailed understanding of such flows has been elusive. While much progress has been made in gaining insights in to the dynamics of shear driven flows, an understanding of the interaction between combustion processes and other interfacial instabilities remains preliminary. In this work, we characterize Rayleigh-Taylor (RT) instability and the shock-driven Richtmyer-Meshkov (RM) instability in the context of combustion. The vast catalogue of research on non-reacting RT and RM flows has demonstrated these instabilities can be manipulated to achieve more efficient and aggressive mixing in comparison with the canonical Kelvin-Helmholtz (KH) problem. This has motivated our efforts to understand RT/RM instability development in the presence of chemical reactions leading to combustion and heat release – we present results from carefully designed numerical simulations of such flows and identify opportunities and challenges in this research space.