It is demonstrated that deep circulation mixing below the base of the standard convective envelope, and the consequent "cool bottom processing" (CBP) of the CNO isotopes, can reproduce the trend with stellar mass of the C-12/C-13 observations in low mass red giants. (This trend is opposite to what is expected from standard first dredge-up.) Our models assume that extra mixing always reaches to the same distance in temperature from the H-burning shell, and that CBP begins when the H-burning shell erases the molecular weight discontinuity ("mu-barrier") established by first dredge-up. For Pop I stars, none of the other CNO isotopes except N-15 are expected to be altered by CBP. (If O-18 depletion occurs on the AGB, as some observations suggest, it would require that extra mixing reach closer to the H-burning shell on the AGB than on the RGB --- and should also result in a much lower C-12/C-13 ratio than is observed.) CBP increases as one reduces the stellar mass or metallicity --- roughly as 1/M^2 on the RGB, due to the longer RGB of low mass stars, and as 1/Z, due to higher H-shell burning temperatures at low metallicity. In low mass Pop II stars, all the CNO isotopes are expected to be altered by CBP. Field Pop II stars exhibit RGB abundances consistent with the predictions of our CBP models that have been normalized to reproduce the Pop I RGB abundances, but globular cluster stars are observed to encounter much more extensive processing; also, CBP is observed to start near the base of the globular cluster RGB (overcoming any "mu-barrier"). Standard first and second dredge-up are also presented, and enrichment of the interstellar medium, relative to SN. For light elements, see astro-ph/9512122.