  The observed dynamics of gas and stars on galactic and larger scales cannot
be accounted for by self-gravity, indicating that there are large quantities of
unseen matter, or that gravity is non-Newtonian in these regimes. Milgrom's
MOdified Newtonian Dynamics (MOND) postulates that Newton's laws are modified
at very low acceleration, and can account for the rotation curves of galaxies
and some other astrophysical observations, without dark matter. Here we apply
MOND to two independent physical systems: Ly-alpha absorbers and galaxy
clusters. While physically distinct, both are simple hydrodynamical systems
with characteristic accelerations in the MOND regime. We find that Ly-alpha
absorbers are somewhat smaller than in Newtonian gravity with dark matter, but
the result depends crucially on the (unknown) background acceleration field in
which they are embedded. In clusters MOND appears to explain the observed
(baryonic) mass-temperature relation. However, given observed gas density and
enclosed mass profiles and the assumption of hydrostatic equilibrium, MOND
predicts radial temperature profiles which disagree badly with observations. We
show this explicitly for the Virgo, Abell 2199 and Coma clusters, but the
results are general, and seem very difficult to avoid. If this discrepancy is
to be resolved by positing additional (presumably baryonic) dark matter, then
this dark matter must have ~1-3 times the cluster gas mass within 1 Mpc. This
result strongly disfavors MOND as an alternative to dark matter (Abridged).
