  We derive sharp Adams inequalities for the Riesz and other potentials of
functions with arbitrary compact support in R^n. Up to now such results were
only known for a class of functions whose supports have uniformly bounded
measure. We obtain several sharp Moser-Trudinger inequalities for the critical
Sobolev space W^{d,n/d} on R^n and on the hyperbolic space H^n The only known
results so far are for d= 1, both on R^n and H^n, and for d = 2 on R^n. Other
sharp inequalities are obtained for general elliptic operators with constant
coefficients and for trace type Borel measures. We introduce critical potential
spaces on which our results can be extended to noninteger values of d.
