  Asymptotic cones of metric spaces were first invented by Gromov. They are
metric spaces which capture the 'large-scale structure' of the underlying
metric space. Later, van den Dries and Wilkie gave a more general construction
of asymptotic cones using ultrapowers. Certain facts about asymptotic cones,
like the completeness of the metric space, now follow rather easily from
saturation properties of ultrapowers, and in this survey, we want to present
two applications of the van den Dries-Wilkie approach. Using ultrapowers we
obtain an explicit description of the asymptotic cone of a semisimple Lie
group. From this description, using semi-algebraic groups and non-standard
methods, we can give a short proof of the Margulis Conjecture. In a second
application, we use set theory to answer a question of Gromov.
