The experiments of enzyme kinetics are measurements of product formed versus time. In the past, such measurements have been converted to the first derivative and extrapolated to zero time, and analyzed in terms of the initial rate versus the initial substrate concentration. Extrapolating derivatives is widely recognized not to be the method of choice, simply one made necessary by the lack of adequate theoretical expressions for product versus time. I have, over the last two years, obtained integrated rate equations for enzyme catalyzed first- and second-order reactions, both reversible and irreversible. An a priori choice of mechanism is not required. The equations are much less complex than the results of previous attempts at integration. In particular, the equations for irreversible reactions appear to be directly suitable for experimental analysis; preliminary experiments in my laboratory show that this is true for a reaction with the stoichiometry A -- P + Q. I propose to develop the experimental and analytical techniques needed to determine the kinetic constants of irreversible second-order reactions using integrated rate equations, and to begin on reversible reactions. I intend to extend my theoretical work to third-order reactions and, if possible, begin experimental work on a simple one. Development of suitable computer programs should ultimately mean that kinetic analysis by means of integrated rate equations is no more complex than, and considerably more efficient than, kinetic analysis by initial rate studies. In fundamental terms, this work will mean that the quantitative behavior of an enzyme can be characterized over the full time course, rather than at a single instant; this much more nearly resembles actual catalytic conditions. It will also ultimately make possible experiments to determine the limits of the steady state assumption.