Little is known of the detailed mechanisms of the polymerization reactions carried out by RNA and DNA polymerases. Besides technical reasons, there are mathematical difficulties not encountered in traditional enzymology. The product of the reaction after one polymerization step is also the substrate of the next step. A number of polymerases, isolated from various sources, have an exonuclease activity. The chain which is being synthesized may be either elongated or trimmed, and its growth has the character of a random walk. In this case, although the overall reaction scheme is more complex, the experiments are more informative, as every dNTP may be transformed into two distinct products: incorporated, or free dNMP. Having solved some of the mathematical difficulties of the random walk problem, we are able to propose a strategy for the study of the polymerization/excision kinetics. We measure the amount y(t) of nucleotide that is polymerized at time tand the amount x(t) of nucleoside monophosphate that has accumulated. When dy dx is plotted against the concentration of dNTP, a curve is obtained with a characteristic shape, a straight line in a large number of cases. From there, kinetic constants can be estimated. The analysis is made in terms of four possible kinetic schemes. In the most elementary model there are only two rate constants, one for incorporation and one for excision. This model is a limiting case of all other models. The frayed-unfrayed model of Brutlag & Kornberg (1972), Hopfield's kinetic proofreading scheme (Hopfield, 1974), and the delayed-escape scheme (Ninio, 1975) are examined in detail, and we show how the kinetic experiments may in principle distinguish between the schemes. Our approach is illustrated with three experiments in which Escherichia coli DNA polymerase I acts on poly(dC), and poly(dT) · oligo(dA)10. © 1979.