We construct and study weak solutions to stochastic differential equation $dX(t)=-b(X(t))dt+\sqrt{2}dW(t)$, $X(0)=x \in \mathbb R^d$,$d \geq 3$, with the vector field $b$ having critical-order singularities. More precisely, $b$ is in the class of weakly form-bounded vector fields, containing, as proper subclasses, the class $L^d$, as well as classes of vector fields having critical-order singularities such as the weak $L^d$ class, the Kato class, the Campanato-Morrey class. Joint with Yu.A.Semenov (Toronto).