Summary Homogeneous catalysis can be classified into single-species and complex catalysis, although the distinction is not always clear-cut. In the former, a single molecule or ion acts as the catalyst; in the latter, the catalyst is a system of several species that interconvert into one another and differ in their catalytic properties. A further complication arises if significant fractions of the total catalyst material may be present in the form of reaction intermediates rather than free catalyst. If so, the concentration of the free catalyst is not known and may vary with conversion, and rate equations that instea contain the known, total amount of catalyst material are needed. In single-species catalysis, the rate laws for noncatalytic reactions apply, the only difference being that the catalyst appears as both a reactant and a product. In catalysis by highly concentrated acids, anomalies may appear: Protonated species other than H3O+ (or protonated solvent in non-aqueous media) may arise and act as additional catalysts; this can be accounted for with the Hammett acidity function. Also, the rates of reactions such as hydration and hydrolysis may decrease with further increase in acid concentration because of reduced availability of free water as reactant. In acid-base catalysis, both an acid (or base) and its conjugate base (or acid) take part in different reaction steps and are eventually restored. Such reactions are first order in acid (or base) if the link-up with that species controls the rate, or first order in H+ (or OH−) if a subsequent step involving the conjugate base (or acid) does so. Traditionally, the first alternative is called “general” acid or base catalysis; the second, “specific” acid or base catalysis. However, this distinction is not always applicable as there may be no clear-cut rate-controlling step, and reversibility of later steps may produce a more complex behavior. Many reactions of organic chemistry are catalyzed by complexes of transition-metal ions, most notably those of Group VIII. Here, the catalyst is a system of different complexes that are linked by ligand-exchange and ligand-dissociation equilibria and differ in their catalytic properties, occasionally with rather counterintuitive results such as a decrease in rate with increase in pressure in a reaction in which gas is consumed. For historical interest and to illustrate a general facet of systems with arbitrary distribution of catalyst material over free catalyst and reaction intermediates, the classical models of enzyme catalysis are briefly reviewed. They show “saturation kinetics:” An increase in reactant concentration causes a shift of catalyst material from free catalyst to an intermediate, so that the rate has an asymptotic limit that can at most be approached even at the highest reactant concentrations. A general formula for single catalytic cycles with arbitrary number of members and arbitrary distribution of catalyst material has been derived by Christiansen. Unfortunately, the denominator of his rate equation for a cycle with k members contains k2 additive terms. Such a profusion makes it imperative to reduce complexity. If warranted, this can be done with the concept of relative abundance of catalyst-containing species or the approximations of a rate-controlling step, quasi-equilibrium steps, or irreversible steps, or combinations of these (the Bodenstein approximation of quasi-stationary states is already implicit in Christiansen's mathematics). In some fortunate instances, the rate equation reduces to a simple power law. If significant fractions of the catalyst material may be bound in the form of reaction intermediates, the rules for reaction orders in noncatalytic simple pathways no longer apply. However, if one of the cycle members—the free catalyst or an intermediate—is a macs (most abundant catalyst-containing species, containing practically all of the catalyst material), the rules for noncatalytic pathways can be adapted: The rate equation and reaction orders for the cycle are the same as for an imaginary equivalent linear pathway that starts and ends with macs. A cycle member that contains only an insignificant fraction of catalyst material is a lacs (low-abundance catalyst containing species), and the denominator terms it contributes can be dropped. In practice, catalytic networks often involve more than a single cycle. In a very common type of network, a linear pathway is attached to the cycle. This is so, for instance, in reactions with ligand-deficient catalysts. Here, the coordinatively saturated “catalyst” has no activity but, rather, must first lose one of its ligands to provide access for a reactant. Other examples of cycles with attached pathways include systems with inhibition, activation, decay, and poisoning. Also, the network may consist of two or more cycles with a common member or pathway. This situation is typical for reactions yielding different isomeric products. The Christiansen formula is extended to cover such cases. In some reactions, the rate increases rather than decreases as conversion progresses. This is loosely called autocatalysis, although no genuine catalysis may be involved. The acceleration may stem from promotion by a product or major early intermediate, or from consumption of a reactant that functions as inhibitor. In product-promoted reactions, the kinetics order with respect to a product (or early intermediate) is positive. This causes the rate to increase to a maximum and then to decline as the effect of consumption of the reactant or reactants begins to overcompensate that of promotion by the product. In reactant-inhibited reactions, the order with respect to a reactant is negative. The rate may increase until the respective reactant is used up and, in some cases, may theoretically approach infinity. The latter behavior is, of course, physically impossible, and another mechanism or event necessarily takes over. A number of the concepts used have parallels in heterogeneous catalysis. For context, the analogies are briefly reviewed. Examples include acetal hydrolysis, base-catalyzed aldol condensation, olefin hydroformylation catalyzed by phosphine-substituted cobalt hydrocarbonyls, phosphate transfer in biological systems, enzymatic transamination, adiponitrile synthesis via hydrocyanation, olefin hydrogenation with Wilkinson's catalyst, and osmium tetroxide-catalyzed asymmetric dihydroxylation of olefins.