It is difficult if not impossible to synthesize long polymers of amino acids, nucleotides, etc., in homogeneous aqueous solution. We suggest that long polymers were synthesized on the surface of minerals in a prebiotic process analogous to solid-phase synthesis. Provided that the affinity of a mineral for an oligomer increases with the length of the oligomer, adsorption must become essentially irreversible for sufficiently long oligomers. Irreversibly adsorbed oligomers may be elongated indefinitely by repeated cycles in which the mineral with its adsorbed oligomers is first incubated with activated monomers and then washed free of deactivated monomer and side-products. We discuss in some detail the formation of oligomers of negatively-charged amino acids such as glutamic acid on anion-exchange minerals such as hydroxylapatite or illite. We show that the average length of adsorbed oligomers at steady state, (n) over bar, depends on the balance between the rate of chain elongation and the rate of hydrolysis, and we derive a very approximate formula for (n) over bar.