  The COBE detection of microwave anisotropies provides the best way of fixing
the amplitude of cosmological fluctuations on the largest scales. We discuss
the impact of this new, precise normalization and give fitting formulae for the
horizon-crossing amplitude as a function of \Omega_0 and n for both open and
flat cosmologies. We also discuss the relevant normalization (\sigma_8) at
galaxy-clustering scales. Already it is clear that the inferred \sigma_8 can be
unnaccepatably high for some of the simplest inflationary models, although many
minor variants give an adequate fit. Generic topological defect models appear
to fare rather badly, and it is unclear whether minor variants or improved
calculations will help much. The detection and mapping of structure in the CMB
anisotropy spectrum on smaller scales in the near future will enable us to
achieve much stronger constraints on models.
