  We determine square root domains for non-self-adjoint Sturm--Liouville
operators of the type $$ L_{p,q,r,s} = -
\frac{d}{dx}p\frac{d}{dx}+r\frac{d}{dx}-\frac{d}{dx}s+q $$ in $L^2((c,d);dx)$,
where either $(c,d)$ coincides with the real line $\mathbb{R}$, the half-line
$(a,\infty)$, $a \in \mathbb{R}$, or with the bounded interval $(a,b) \subset
\mathbb{R}$, under very general conditions on the coefficients $q, r, s$. We
treat Dirichlet and Neumann boundary conditions at $a$ in the half-line case,
and Dirichlet and/or Neumann boundary conditions at $a,b$ in the final interval
context. (In the particular case $p=1$ a.e.\ on $(a,b)$, we treat all separated
boundary conditions at $a, b$.)
