  For every genus $g$, we prove that $S^2 \times R$ contains complete, properly
embedded, genus-$g$ minimal surfaces whose two ends are asymptotic to helicoids
of any prescribed pitch. We also show that as the radius of the $S^2$ tends to
infinity, these examples converge smoothly to complete, properly embedded
minimal surfaces in $R^3$ that are helicoidal at infinity. We prove that
helicoidal surfaces in $R^3$ of every prescribed genus occur as such limits of
examples in $S^2\times R$.
