  We analyse a multilevel Monte Carlo method for the approximation of
distribution functions of univariate random variables. Since, by assumption,
the target distribution is not known explicitly, approximations have to be
used. We provide an asymptotic analysis of the error and the cost of the
algorithm. Furthermore we construct an adaptive version of the algorithm that
does not require any a priori knowledge on weak or strong convergence rates. We
apply the adaptive algorithm to smooth path-independent and path-dependent
functionals and to stopped exit times of SDEs.
