  In gaussian theories of structure formation, the galaxy cluster abundance is
an extremely sensitive probe of the density fluctuation power spectrum and of
the density parameter, $\Omega$. We develop this theme by deriving and studying
in detail the mass function of collapsed objects and its relation to these
quantities. Application to current data yields constraints which are degenerate
between the amplitude of the perturbations and the density parameter; we
nevertheless obtain an important limit on the present day mass perturbation
amplitude as a function of $\Omega$ and can rule-out the `standard' cold dark
matter (CDM) model. Future observations of the evolution of the cluster
abundance will break the degeneracy and provide important constraints on both
the power spectrum and the density parameter, individually. We focus primarily
on X-ray clusters in the discussion, and finish with a presentation of the
promising new field of Sunyaev-Zel'dovich observations of cluster evolution.
