The fact that the simplest modern cosmological theory, standard Cold Dark Matter (sCDM), almost fits all available data has encouraged the search for variants of CDM that can do better. Here we discuss what are perhaps the two most popular variants of CDM that might agree with the data: ACDM and CHDM. While the predictions of COBE-normalized ACDM and CHDM both agree well with the available data on scales of ~ 10 to 100 h−1 Mpc, each has potential virtues and defects. ACDM with Ω0 ~ 0.3 has the possible virtue of allowing a higher expansion rate H0 for a given cosmic age t0, but the defect of predicting too much fluctuation power on small scales. CHDM has less power on small scales, so its predictions appear to be in good agreement with data on the galaxy distribution, but it remains to be seen whether it predicts early enough galaxy formation to be compatible with the latest high-redshift data. Also, several sorts of data suggest that neutrinos have nonzero mass. And two very recent observational results favor high cosmic density, and thus favor Ω = 1 models such as CHDM over ACDM — (1) the positive deceleration parameter q0 > 0 measured using high-redshift Type Ia supernovae, and (2) the low primordial deuterium/hydrogen ratio measured in two different quasar absorption spectra. If confirmed, (1) rules out a cosmological constant large enough to help significantly with the H0-t0 problem; while (2) suggests that the baryonic cosmological density is at the upper end of the range allowed by Big Bang Nucleosynthesis, perhaps high enough to resolve the “cluster baryon crisis” for Ω = 1 models. We try to identify “best” variants of both ACDM and CHDM, and discuss critical observational tests for both models.