  We derive sharp strong convergence rates for the Euler-Maruyama scheme
approximating multidimensional SDEs with multiplicative noise without imposing
any regularity condition on the drift coefficient. In case the noise is
additive, we show that Sobolev regularity can be leveraged to obtain improved
rate: drifts with regularity of order $\alpha \in (0,1)$ lead to rate
$(1+\alpha)/2$.

Probability