We present very deep spectrophotometry of 14 bright extragalactic H II regions belonging to spiral, irregular, and blue compact galaxies. The data for 13 objects were taken with the High Resolution Echelle Spectrometer on the Keck I telescope. We have measured C II recombination lines in 10 of the objects and O II recombination lines in eight of them. We have determined electron temperatures from line ratios of several ions, especially those of low ionization potential. We have found a rather tight linear empirical relation between T e([N II]) and T e([O III]). We have found that O II lines give always larger abundances than [O III] lines. Moreover, the difference of both O++ abundance determinations—the so-called abundance discrepancy factor—is very similar in all the objects, with a mean value of 0.26 ± 0.09 dex, independent of the properties of the H II region and of the parent galaxy. Using the observed recombination lines, we have determined the O, C, and C/O radial abundance gradients for three spiral galaxies: M33, M101, and NGC 2403, finding that C abundance gradients are always steeper than those of O, producing negative C/O gradients across the galactic disks. This result is similar to that found in the Milky Way and has important implications for chemical evolution models and the nucleosynthesis of C. Most of the data presented herein were obtained at the W.M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W.M. Keck Foundation. Part of the observations were made with the 4.2 m William Herschel Telescope (WHT), operated on the island of La Palma by the Isaac Newton Group in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias.