NADH:ubiquinone oxidoreductase (complex I) is the largest and most complicated enzyme in the mitochondrial electron transfer chain. It catalyses the oxidation of NADH and the reduction of ubiquinone, coupled to the translocation of protons across the mitochondrial inner membrane, maintaining the proton motive force used for ATP synthesis. Complex I is the least understood of the respiratory enzymes; although the mechanisms of NADH oxidation and intramolecular electron transfer are gradually becoming appreciated, the mechanisms of quinone binding and reduction and proton translocation remain unknown. Complex I dysfunction has been implicated in a wide range of pathologies including mitochondrial diseases such as Leigh's disease, as well as neurodegenerative diseases such as Alzheimer's and Parkinson's. The work described in the first part of this thesis is aimed at elucidating the structure of either a subcomplex of mitochondrial complex I, or of the intact enzyme itself. A comprehensive investigation revealed that hydrophilic subcomplexes of complex I from bovine heart mitochondria are not suitable for use as models of the intact enzyme. Attempts to prepare intact complex I of sufficient quality for structural work were successful; however, results from a large set of crystallization trials were disappointing. The second part of this thesis describes three studies of the function and mechanism of complex I from bovine heart mitochondria. First, the flavin mononucleotide, the site of NADH oxidation, was identified as the site of the 'inhibitor-insensitive' NADH:ubiquinone oxidoreduction reaction. The formation of semiquinones initiates redox cycling reactions with oxygen, producing vast amounts of reactive oxygen species; further studies revealed that other oxidants, such as paraquat, also react at the flavin site and initiate redox cycling reactions. Second, kinetic studies showed that the reaction between NADH and positively charged oxidants such as HAR (hexaammineruthenium (III)) proceeds by an unusual ternary reaction mechanism at the flavin site of complex I. Finally, double electron-electron resonance spectroscopy was used to show unambiguously that iron sulphur cluster 4Fe[TY]1 gives rise to electron paramagnetic resonance signal N4; the data provide an alternating potential energy profile for electron transfer along the cluster chain between the flavin and the quinone-binding site.