We investigate the physical basis, validity, and limitations of the minimum electrophilicity principle, MEP, which postulates that the sum of the electrophilicity indices, Sigma omega, of the reaction products will be smaller than that of the reactants, Delta omega < 0. We present a much-improved understanding of the conditions for minimizing electrophilicity indices. Two indices, omega(1) = (I + A)(2)/ 8 (I - A) and omega(2) = I.A/ (I - A), are discussed, using ionization energies, I, and electron affinities, A, obtained from either ground-state (GS) or valence-state (VS) energies. The performances of omega(1 )and omega(2) are compared for a wide range of chemical species from diatomic molecules, through large clusters to liquid water and solid crystals. New analytical arguments in support of MEP are found. Two new theorems are proved, and three new rules rationalize the changes Delta omega(1) and Delta omega(2) in association reactions, X + Y -> XY. They explain why MEP is much more successful as a guiding rule than the maximum hardness postulate in such reactions. On the other hand, they also identify the increased electron affinity of the product as the reason for the rare but highly significant failures of MEP, e.g., in B-2, C-2, Si-2, and CN. As a rule, electrophilicity is minimized in association reactions. However, both omega(1 )and omega(2) are increased if the bond dissociation energy D(XY-) is larger than D(XY), which is equivalent to an increased product electron affinity. The large positive changes Delta omega(1) and Delta omega(2) in 2C -> C-2 exhibit a strong contrast to MEP. The changes in electrophilicity indices may help gain insights into the versatility of the chemistries of carbon and other elements. Solid-state double-exchange reactions are correctly assessed by Kaya's composite descriptor, somewhat less by omega(2), but not at all by omega(1). A wide class of failures of MEP is found as sizedriven electrophilicity maximization, Delta omega > 0, e.g., in fullerenes, large metal clusters, and liquid water. Many electrophiles, especially superelectrophiles, show significantly larger electrophilicity indices than the largest index of their isolated atoms. The changes Delta omega(1) and Delta omega(2) provide important information on the reactivities of chemical systems; however, it appears that the minimum electrophilicity postulate cannot serve as a basis for a theory.