  This paper provides conditions under which subsampling and the bootstrap can
be used to construct estimators of the quantiles of the distribution of a root
that behave well uniformly over a large class of distributions $\mathbf{P}$.
These results are then applied (i) to construct confidence regions that behave
well uniformly over $\mathbf{P}$ in the sense that the coverage probability
tends to at least the nominal level uniformly over $\mathbf{P}$ and (ii) to
construct tests that behave well uniformly over $\mathbf{P}$ in the sense that
the size tends to no greater than the nominal level uniformly over
$\mathbf{P}$. Without these stronger notions of convergence, the asymptotic
approximations to the coverage probability or size may be poor, even in very
large samples. Specific applications include the multivariate mean, testing
moment inequalities, multiple testing, the empirical process and U-statistics.
