A grid of models for cool dwarf and giant stars has been used to calculate synthetic spectra in the region of the 2.3-mu-m CO bands. These synthetic spectra have been compared with observed spectra for four stars with known abundances, effective temperatures, and surface gravities. The carbon abundances and carbon isotope ratios found from this comparison agree well with those found from high dispersion spectroscopy. The synthetic spectra have been used to calculate tables of CO indices on two different systems. The observed dependence of CO band strength on luminosity is seen in these synthetic indices. These CO indices have also been used to derive carbon abundances for fifteen field giant stars with known atmospheric parameters. The resulting carbon abundances are again in good agreement with those found from high dispersion spectroscopy. Sample CO spectra are presented for globular cluster giant branches. Calculations of CO colors for different oxygen abundances and with carbon abundances which fit the observed G bands suggest that the oxygen abundance is not enhanced in the red giant stars in the very metal-poor cluster NGC 6397.