  We consider upper bounds for the approximation error E|g(X)-g(\hat X)|^p,
where X and \hat X are random variables such that \hat X is an approximation of
X in the L_p-norm, and the function g belongs to certain function classes,
which contain e.g. functions of bounded variation. We apply the results to the
approximations of a solution of a stochastic differential equation at time T by
the Euler and Milstein schemes. For the Euler scheme we provide also a lower
bound.
