  We investigate the validity and failure of Liouville theorems and Harnack
inequalities for parabolic and elliptic operators with low regularity
coefficients. We are particularly interested in operators of the form
$\partial_t - \Delta +b\cdot\nabla$ and $-\Delta +b\cdot\nabla$ with a
divergence-free drift $b$. We prove the Liouville theorem and Harnack
inequality when $b\in L_\infty(BMO^{-1})$ resp. $b\in BMO^{-1}$ and provide a
counterexample to such results demonstrating sharpness of our conditions on the
drift. Our results generalize to divergence-form operators with an elliptic
symmetric part and a BMO skew-symmetric part. We also prove the existence of a
modulus of continuity for solutions to the elliptic problem in two dimensions,
depending on the non-scale-invariant norm $\|b\|_{L_1}$. In three dimensions,
on the other hand, bounded solutions with $L_1$ drifts may be discontinuous.
