  Consider stochastic differential equations (SDEs) in $\Rd$:
$dX_t=dW_t+b(t,X_t)\d t$, where $W$ is a Brownian motion, $b(\cdot, \cdot)$ is
a measurable vector field. It is known that if $|b|^2(\cdot,
\cdot)=|b|^2(\cdot)$ belongs to the Kato class $\K_{d,2}$, then there is a weak
solution to the SDE. In this article we show that if $|b|^2$ belongs to the
Kato class $\K_{d,\a}$ for some $\a \in (0,2)$ ($\a$ can be arbitrarily close
to $2$), then there exists a unique strong solution to the stochastic
differential equations, extending the results in the existing literature as
demonstrated by examples. Furthermore, we allow the drift to be time-dependent.
The new regularity estimates we established for the solutions of parabolic
equations with Kato class coefficients play a crucial role.
