Let $ℒ=$ -div $\left(A\right(x\left)\nabla \right)$ be a second order elliptic operator with real, symmetric, bounded measurable coefficients on ${ℝ}^n$ or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed $p\>2$, a necessary and sufficient condition is obtained for the boundedness of the Riesz transform $\nabla {\left(ℒ\right)}^{-1/2}$ on the $L^p$ space. As an application, for $1\<p\<3+ϵ$, we establish the $L^p$ boundedness of Riesz transforms on Lipschitz domains for operators with $VMO$ coefficients. The range of $p$ is sharp. The closely related boundedness of ...