For accurate thermochemical tests of electronic structure theory, accurate true anharmonic zero-point vibrational energies ZPVE(true) are needed. We discuss several possibilities to extract this information for molecules from density functional or wave function calculations and/or available experimental data: (1) Empirical universal scaling of density-functional-calculated harmonic ZPVE(harm)s, where we find that polyatomics require smaller scaling factors than diatomics. (2) Direct density-functional calculation by anharmonic second-order perturbation theory PT2. (3) Weighted averages of harmonic ZPVE(harm) and fundamental ZPVE(fund) (from fundamental vibrational transition frequencies), with weights (3/4, 1/4) for diatomics and (5/8, 3/8) for polyatomics. (4) Experimental correction of the PT2 harmonic contribution, i.e., the estimate ZPVE(PT2)(true) + (ZPVE(expt)(fund) -ZPVE(PT2)(fund)) for ZPVE(true). The (5/8, 3/8) average of method 3 and the additive correction of method 4 have been proposed here. For our database of experimental ZPVE(true), consisting of 27 diatomics and 8 polyatomics, we find that methods 1 and 2, applied to the popular B3LYP and the nonempirical PBE and TPSS functionals and their one-parameter hybrids, yield polyatomic errors on the order of 0.1 kcal/mol. Larger errors are expected for molecules larger than those in our database. Method 3 yields errors on the order of 0.02 kcal/mol, but requires very accurate (e.g., experimental, coupled cluster, or best-performing density functional) input harmonic ZPVE(harm). Method 4 is the best-founded one that meets the requirements of high accuracy and practicality, requiring as experimental input only the highly accurate and widely available ZPVE(expt)(fund) and producing errors on the order of 0.05 kcal/mol that are relatively independent of functional and basis set. As a part of our study, we also test the ability of the density functionals to predict accurate equilibrium bond lengths and angles for a data set of 21 mostly polyatomic molecules (since all calculated ZPVEs are evaluated at the correspondingly calculated molecular geometries).