A number of density functional (DF) methodologies were systematically examined for their ability to describe strong hydrogen bonds. A set of 10 ion/molecule systems, 5 cationic and 5 anionic, was chosen on the basis of the availability of experimental and high-level ab initio results against which the DF methods could be compared. All DF models used the Lee-Yang-Parr (LYP) functional for nonlocal electron correlation, and either the 1988 Becke (B) or 1993 Becke "three-parameter" (B3) nonlocal exchange functional. Full geometry optimizations were carried out with the B-LYP combination and Gaussian basis sets, including Pople bases from 6-31G(d,p) to 6-311++G(d,p), and the TZVP and TZVP+ bases, as well as with the numerical DMol DNP and DNPP basis sets, but B3-LYP was used only with 6-311++G(d,p). Vibrational modes and thermodynamic functions were calculated only with the B-LYP/ and B3-LYP/6-311++G(d,p) models. The results support the following major conclusions. Nonlocal DF models require diffuse atomic functions to adequately describe strong H-bonding systems, especially in anionic complexes. With 6-311++G(d,p) the B-LYP and B3-LYP models are slightly inferior, but still quite similar, to MP2. B3-LYP is clearly the better DF model, yielding complexation enthalpies with root mean square deviations of 0.7 and 1.6 kcal/mol, respectively, from the MP2 and experimental values. More satisfactory agreement with experiment can be obtained in individual cases by judicious enlargement of the basis set. Efficient auxiliary basis/model density DF methods are much faster than ex post facto DF schemes based on existing ab initio methodology, which in turn are appreciably faster than MP2 with the same basis set. DF models of good quality will extend to strong H-bonded systems large enough to be of direct bioorganic relevance.