The costs in computer time and memory are formidable for numerical simulations of reacting flows. Major costs are in integrating the ordinary differential equations describing the chemical reactions, evaluating diffusive transport processes and in the solution to the convective transport when there are irregular boundaries or obstacles. Dealing with these problems requires improving numerical algorithms and other software, and taking advantage of larger, faster, more accurate and friendlier computers. Over the last few decades, improved algorithms have contributed to simulation capability at least as much as hardware development. Together, hardware and software advances have increased our capability by orders of magnitude. Early one-dimensional gas-dynamics calculations were resolved with 40–50 computational cells. Such problems can now be performed in three dimensions, about 4–5 orders of magnitude more computer time, in the same time as the previous one-dimensional calculations. Here we discuss the forefront developments in computers and computational approaches, current limitations on the types of detailed computations of reactive flows, the use of phenomenological models to replace portions of the calculation, and conclude with two examples of complex reactive flow calculations.