Atomization energies at 0 K and heats of formation at 0 and 298 K are predicted for the neutral and ionic NxFy and OxFy systems using coupled cluster theory with single and double excitations and including a perturbative triples correction (CCSD(T)) method with correlation consistent basis sets extrapolated to the complete basis set (CBS) limit. To achieve near chemical accuracy (+/- 1 kcal/mol), three corrections to the electronic energy were added to the frozen core CCSD(T)/CBS binding energies: corrections for core valence, scalar relativistic, and first order atomic spin orbit effects. Vibrational zero point energies were computed at the CCSD(T) level of theory where possible. The calculated heats of formation are in good agreement with the available experimental values, except for FOOF because of the neglect of higher order correlation corrections. The F+ affinity in the NxFy series increases from N-2 to N2F4 by 63 kcal/mol, while that in the O2Fy series decreases by 18 kcal/mol from O-2 to O2F2. Neither N-2 nor N2F4 is predicted to bind F- and N2F2 is a very weak Lewis acid with an F- affinity of about 10 kcal/mol for either the cis or trans isomer. The low F- affinities of the nitrogen fluorides explain why, in spite of the fact that many stable nitrogen fluoride cations are known, no nitrogen fluoride anions have been isolated so far. For example, the F- affinity of NF is predicted to be only 12.5 kcal/mol which explains the numerous experimental failures to prepare NF2- salts from the well-known strong acid HNF2. The F- affinity of O-2 is predicted to have a small positive value and increases for O2F2 by 23 kcal/mol, indicating that the O2F3- anion might be marginally stable at subambient temperatures. The calculated adiabatic ionization potentials and electron affinities are in good agreement with experiment considering that many of the experimental values are for vertical processes.