Abstract <p> The present paper provides a survey of the achievements of the research group led by Professor N. N. Nefedov in the field of asymptotic analysis of moving fronts for nonlinear singularly perturbed reaction–diffusion–advection equations. The primary research tool is the asymptotic method of differential inequalities, which allows one to carry out a rigorous justification of the proposed asymptotic approximations. The paper systematizes results for parabolic initial-boundary value problems and illustrates key approaches to their analysis. </p>