  Hadron-quark combustion in dense matter is a central topic in the study of
phases in compact stars and their high-energy astrophysics. We critically
review the literature on hadron-quark combustion, dividing them into a "first
wave" that treats the problem as a steady-state burning with or without
constraints of mechanical equilibrium, and a "second wave" which uses numerical
techniques to resolve the burning front and solves the underlying Partial
Differential Equations for the chemistry of the burning front under less
restrictive conditions. We detail the inaccuracies that the second wave amends
over the first wave, and highlight crucial differences between various
approaches in the second wave. We also include results from time-dependent
simulations of the reaction zone that include a hadronic EOS, neutrinos, and
self-consistent thermodynamics without using parameterized shortcuts.
