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Polymer chain growth that proceeds through a catalyst-mediated cycle—insertion at a metal center, coordination of a monomer, or turnover of an organocatalyst—constitutes catalytic polymerization. The catalyst is regenerated at each addition and is not consumed as a stoichiometric initiator in the living-ionic sense, although a single metal site may remain attached to one chain (single-site catalysis). Neutral nickel(II) complexes for ethylene polymerization exemplify well-defined molecular catalysts whose ligand substituents control activity and regiochemistry ^[1]. Living aqueous catalytic polymerization can produce uniform, monodisperse single-chain nanocrystals when initiation and propagation remain confined to the catalytic site ^[2]. Topology—linear, cyclic, branched, or dendritic—is a direct consequence of how the catalyst binds and releases chain ends ^[3]]. The concept overlaps coordination and olefin polymerization but is broader: catalytic ring-opening of cyclic esters and catalytic step-growth couplings are included whenever a catalytic cycle, rather than a persistent ionic or radical chain end alone, is the growth engine. Uncatalyzed condensation and conventional free-radical polymerization without a controlling catalyst lie outside the term ^[3].