  Our focus is on the design and analysis of efficient Monte Carlo methods for
computing tail probabilities for the suprema of Gaussian random fields, along
with conditional expectations of functionals of the fields given the existence
of excursions above high levels, b. Na\"{i}ve Monte Carlo takes an exponential,
in b, computational cost to estimate these probabilities and conditional
expectations for a prescribed relative accuracy. In contrast, our Monte Carlo
procedures achieve, at worst, polynomial complexity in b, assuming only that
the mean and covariance functions are H\"{o}lder continuous. We also explain
how to fine tune the construction of our procedures in the presence of
additional regularity, such as homogeneity and smoothness, in order to further
improve the efficiency.
