Cross-linking in proteins by alpha,beta-dicarbonyl compounds is one of the most damaging consequences of reactive carbonyl species in vivo and in foodstuffs. In this article we investigate computationally the cross-linking of glyoxal and methylglyoxal with lysine and arginine residues using density functional theory and the wB97XD dispersion-corrected functional. Five pathways, A-E, have been characterized. In pathways A and B, the reaction proceeds via formation of the Schiff base, aldimine, followed by addition of arginine. In contrast, in pathways C-E, direct addition of arginine to the dicarbonyl compounds occurs first, leading to a dihydroxyimidazolidine intermediate, which then reacts with lysine after dehydration and proton transfer reactions. The results reveal that pathways A, C, and E are competitive whereas reactions via pathways B and D are much less favorable. Inclusion of up to five explicit water molecules in the proton transfer and dehydration steps is found to lower the energy barriers in the feasible pathways by about 5-20 kcal/mol. Comparison of the mechanisms of methylglyoxal-derived imidazolium cross-linking (MODIC) and glyoxal-derived imidazolium cross-linking (GODIC) shows that the activation barriers are lower for GODIC than MODIC, in agreement with experimental observations.