The goal of this paper is to study the Riesz transforms $\na A^{-1/2}$ where
$A$ is the Schr\"{o}dinger operator $-\D-V, V\ge 0$, under different conditions
on the potential $V$. We prove that if $V$ is strongly subcritical, $\na
A^{-1/2}$ is bounded on $L^p(\R^N)$, $N\ge3$, for all $p\in(p_0';2]$ where
$p_0'$ is the dual exponent of $p_0$ where $2<\frac{2N}{N-2}<p_0<\i$; and we
give a counterexample to the boundedness on $L^p(\R^N)$ for
$p\in(1;p'_0)\cup(p_{0*};\i)$ where $p_{0*}:=\frac{p_0N}{N+p_0}$ is the reverse
Sobolev exponent of $p_0$. In addition, if the potential is in $L^{N/2}(\R^N)$,
then $\na A^{-1/2}$ is bounded on $L^p(\R^N)$ for all $p\in(p_0';p_{0*})$. If
the potential is strongly subcritical in the Kato subclass $K_N^{\i}$, then
$\na A^{-1/2}$ is bounded on $L^p(\R^N)$ for all $p\in(1;2]$, moreover if it is
in $L^{N/2}(\R^N)$ then $\na A^{-1/2}$ is bounded on $L^p(\R^N)$ for all
$p\in(1;N)$. We prove also boundedness of $V^{1/2}A^{-1/2}$ with the same
conditions on the same spaces.