  This article reviews the application of advanced Monte Carlo techniques in
the context of Multilevel Monte Carlo (MLMC). MLMC is a strategy employed to
compute expectations which can be biased in some sense, for instance, by using
the discretization of a associated probability law. The MLMC approach works
with a hierarchy of biased approximations which become progressively more
accurate and more expensive. Using a telescoping representation of the most
accurate approximation, the method is able to reduce the computational cost for
a given level of error versus i.i.d. sampling from this latter approximation.
All of these ideas originated for cases where exact sampling from couples in
the hierarchy is possible. This article considers the case where such exact
sampling is not currently possible. We consider Markov chain Monte Carlo and
sequential Monte Carlo methods which have been introduced in the literature and
we describe different strategies which facilitate the application of MLMC
within these methods.
