We combine the diffuse basis functions from the 6-31+G basis set of Pople and co-workers with the correlation-consistent basis sets of Dunning and co-workers. In both wave function and density functional calculations, the resulting basis sets reduce the basis set superposition error almost as much as the augmented correlation-consistent basis sets, although they are much smaller. In addition, in density functional calculations the new basis sets, called cc-pVxZ+ where x = D, T, Q, ⋯, or x = D+d, T+d, Q+d, ⋯, give very similar energetic predictions to the much larger aug-cc-pVxZ basis sets. However, energetics calculated from correlated wave function calculations are more slowly convergent with respect to the addition of diffuse functions. We also examined basis sets with the same number and type of functions as the cc-pVxZ+ sets but using the diffuse exponents of the aug-cc-pVxZ basis sets and found very similar performance to cc-pVxZ+; these basis sets are called minimally augmented cc-pVxZ, which we abbreviate as maug-cc-pVxZ.