The study of metalloenzymes that oxidize C-H bonds is hampered by a lack of understanding of how transition metal centers oxidize organic substrates. The goal of this work is to provide new mechanistic paradigms for such reactions. A particular focus will be oxidation by hydrogen atom abstraction, which has been implicated as the key substrate-activating step for a number of important metalloenzymes, including cytochrome P-450, lipoxygenase, and dopamine Beta-hydroxylase. Hydrogen atom abstraction also appears to be a key step in C-H bond oxidation by permanganate and other reagents used in organic chemistry, and is involved in industrially important hydrocarbon oxidations. The current picture of these reactions is that there must be a radical at the active site -- such as an oxo-iron group with radical character at the oxygen as suggested for cytochrome P-450 and bleomycin. We propose a new approach to these reactions, based on the affinity of the active site for a hydrogen atom, in other words the O-H bond strength formed. The very extensive literature on hydrogen atom abstraction reactions is dominated by such discussions of bond strengths, not radical character. The affinity of an active site or reagent for a hydrogen atom can be calculated from its redox potential and pKa, adapting a procedure that is well developed for organic and organometallic compounds. Preliminary studies of oxidations by chromyl chloride and permanganate suggest that this perspective is not only qualitative, providing an explanation for why and how reactions occur, but also quantitative: the rate of hydrogen atom abstraction by CrO2Cl2 or MnO4- can be roughly predicted based on the strength of the O-H bond formed. This prediction is based on the Polanyi relation between the rate of radical reactions and their driving force, a simple treatment that is related to the Marcus theory of electron transfer rates. Further studies of chromium (VI) and permanganate oxidations are proposed to test this hypothesis. Related reactions that occur by initial hydride transfer will also be explored. We predict that a variety of coordination complexes should also be able to oxidize C-H bonds, and studies of copper, iron, nickel, manganese, and ruthenium compounds are described. We will begin with known copper(III) and iron(III) coordination complexes, which should be excellent functional models for the C-H activation step in dopamine Beta- hydroxylase and lipoxygenase. Preliminary results suggest that a copper(III) imine-oxime complex does oxidize substrates by hydrogen atom transfer. Confirmation of our hypothesis will facilitate preparation of better models for these oxidases, and will lead to better understanding and prediction of their selectivity.