The COBE microwave background temperature fluctuations and the abundance of local rich clusters of galaxies provide the two most powerful constraints on cosmological models. When all variants of the standard cold dark matter (CDM) model are subject to the combined constraint, the power spectrum of any model is fixed to $\sim 10%$ accuracy in both the shape and overall amplitude. These constrained models are not expected to differ dramatically in their local large-scale structure properties. However, their evolutionary histories differ, resulting in dramatic differences towards high redshift. We examine in detail six standardized, COBE and cluster normalized CDM models with respect to a large set of independent observations. The observations include correlation function of rich clusters of galaxies, galaxy power spectrum, evolution of rich cluster abundance, gravitational lensing by moderate -to-high redshift clusters, $\lya$ forest, damped $\lya$ systems, high redshift galaxies, reionization of the universe and future CMB experiments. It seems that each of the independent observations examined is or potentially is capable of distinguishing between at least some of the models. The combined power of several or all of these observations is tremendous. Thus, we appear to be on the verge of being able to make dramatic tests of all models in the near future using a rapidly growing set of observations, mostly at moderate to high redshift. Consistency or inconsistency between different observed phenomena on different scales and/or at different epochs with respect to the models will have profound implications for theory of growth of cosmic structure.

ApJ in press (1998), 26 emulateapj pages