  This paper aims at developing a systematic study for the weak rate of
convergence of the Euler-Maruyama scheme for stochastic differential equations
with very irregular drift and constant diffusion coefficients. We apply our
method to obtain the rates of approximation for the expectation of various
non-smooth functionals of both stochastic differential equations and killed
diffusion. We also apply our method to the study of the weak approximation of
reflected stochastic differential equations whose drift is H\"older continuous.
