The Cosmic Background Explorer (COBE) microwave background temperature fluc
tuations and the abundance of local rich clusters of galaxies provide the t
wo most powerful constraints on cosmological models. When all variants of t
he standard cold dark matter (CDM) model are subject to the combined constr
aints, the power spectrum of any model is fixed to similar to 10% accuracy
in both the shape and overall amplitude. These constrained models are not e
xpected to differ dramatically in their local large-scale structure propert
ies. However, their evolutionary histories differ, with the differences bei
ng dramatically larger toward higher redshifts. In particular, it should be
true that any statistical measure that probes a rapidly diminishing tail o
f some distribution should provide a sensitive test at some sufficiently hi
gh redshift, when the objects in question are rare and hence in the tail. W
e examine in detail six standardized, COBE- and cluster-normalized CDM mode
ls with respect to a large set of independent observations. The observation
s include the correlation function of rich clusters of galaxies, the galaxy
power spectrum, the evolution of the rich cluster abundance, the gravitati
onal lensing by moderate- to high-redshift clusters, the Ly alpha forest, t
he damped Ly alpha systems, the high-redshift galaxies, the reionization of
the universe, and future cosmic microwave background experiments. It seems
that each of the independent observations examined is potentially capable
of distinguishing among at least some of the models. The combined power of
several or all of these observations is tremendous. Thus, we appear to be o
n the verge of being able to make dramatic tests of all the models in the n
ear future using a rapidly growing set of observations, mostly at moderate
to high redshift. Consistency or inconsistency among different observed phe
nomena on different scales and/or at different epochs with respect to the m
odels will have profound implications for the theory of the growth of cosmi
c structure.