  We analyze the behavior of supercooled fluids under shear both theoretically
and numerically. Theoretically, we generalize the mode-coupling theory of
supercooled fluids to systems under stationary shear flow. Our starting point
is the set of generalized fluctuating hydrodynamic equations with a convection
term. A nonlinear integro-differential equation for the intermediate scattering
function is constructed. This theory is applied to a two-dimensional colloidal
suspension. The shear rate dependence of the intermediate scattering function
and the shear viscosity is analyzed. We have also performed extensive numerical
simulations of a two-dimensional binary liquid with soft-core interactions
near, but above, the glass transition temperature. Both theoretical and
numerical results show: (i) A drastic reduction of the structural relaxation
time and the shear viscosity due to shear. Both the structural relaxation time
and the viscosity decrease as $\dot{\gamma}^{-\nu}$ with an exponent $\nu \leq
1$, where $\dot{\gamma}$ is the shear rate. (ii) Almost isotropic dynamics
regardless of the strength of the anisotropic shear flow.
