  We introduce a new family of real simple modules over the quantum affine
algebras, called the affine determinantial modules, which contains the
Kirillov-Reshetikhin (KR)-modules as a special subfamily, and then prove
T-systems among them which generalize the T-systems among KR-modules and
unipotent quantum minors in the quantum unipotent coordinate algebras
simultaneously. We develop new combinatorial tools: admissible chains of
i-boxes which produce commuting families of affine determinantial modules, and
box moves which describe the T-system in a combinatorial way. Using these
results, we prove that various module categories over the quantum affine
algebras provide monoidal categorifications of cluster algebras. As special
cases, Hernandez-Leclerc categories provide monoidal categorifications of the
cluster algebras for an arbitrary quantum affine algebra.
