Motivated by the study of spray combustion, this work addresses the combustion of non-spherical droplets. The combustion of spray is usually understood through the theory of droplet combustion, and improving this latter theory is the narrow aim of this work. The current work uses perturbation theory to derive a novel model for the vaporization of non-spherical droplets. Compared to previous efforts in this area, the work uses a physics-based approach by incorporating ideas from the asymptotic analysis of Taylor and Acrivos [J. Fluid Mech., 1964]. The perturbation strategy expands the droplet shape using spherical harmonics, and the theory characterizes the shape of the droplet via the Weber number. The introduction of this parameter is key as it is a parameter that can be easily measured in experiments, and thus it can be used to connect the theoretical results with application. The perturbation analysis is performed based around the classical solution of spherical droplet combustion in quiescent flow. The theory indicates that the effect of droplet deformation can be accounted for by a correction to the droplet combustion rate that is simple polynomial function of the droplet Weber number. Results are compared to existing literature, and it confirms the established trend that deformed droplets vaporize faster than spherical droplets. Analysis of the flame shape reveals that the flame remains nearly spherical, however, the mean flame standoff changes with droplet shape. The extension of the theory to high Reynolds number conditions is briefly discussed.