We present a systematic theory of polymer reaction kinetics at an interface separating two immiscible melts, A and B, in each of which a fraction of chains carry reactive end groups. We consider arbitrary values of local group reactivity, Q(b), and reactive group densities in either bulk, n(A)(infinity) and n(B)(infinity), with the convention n(A)(infinity)less than or equal to n(B)(infinity). At short times reaction kinetics are second order in bulk densities. Initially, kinetics are of simple mean field type, with surface density of reaction product after time t given by R-t approximate to Q(b)ha(3)tn(A)(infinity)n(B)(infinity) where h is the interface width and a reactive group size. If Q(b) exceeds a density-dependent threshold a transition occurs, at a time less than the longest polymer relaxation time tau, to second order diffusion-controlled (DC) kinetics with R-t approximate to x(t)(4)n(A)(infinity)n(B)(infinity). Here x(t) is the rms monomer displacement. Logarithmic corrections arise in marginal cases. This leads to R-t similar to t/(ln t) for unentangled chains, while for entangled melts consecutive regimes R-t similar to t/(ln t), R-t similar to t(1/2) and R-t similar to t/(ln t) exist. Which regimes are realized depends on Q(b) and n(B)(infinity). At long times, a transition occurs to first-order DC kinetics. The reaction rate, R-t approximate to x(t)n(A)(infinity), is determined by the more dilute A side, where a density depletion hole of size x(t) develops at the interface. For high reactive chain densities on the B side (n(B)(infinity)R(3) > 1 where R is polymer coil size), and for Q(b) sufficiently large, these kinetics onset before tau. Then R-t similar to t(1/4) for unentangled melts, while for entangled cases consecutive regimes R-t similar to t(1/4), R-t similar to t(1/8), and R-t similar to t(1/4) arise, some or all of which may be realized depending on Q(b) and n(B)(infinity). The final first-order regime is always governed by center of gravity diffusion, R-t similar to t(1/2). At a certain time scale the interface saturates with AB copolymer product and reactions are strongly suppressed. This prevents the onset of the long time first-order DC regime if the reactivity is very small, Q(b) < Q(b)dagger with Q(b)dagger similar to 1/N-1/2 (unentangled melts) or Q(b)dagger similar to 1/N-3/2 (entangled).