For every genus g, we prove that S^2 × 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.