The properties of dilute and semidilute solutions of rigid polyelectrolytes are investigated using integral equation theory. The theory predicts liquidlike structure on length scales of the order of the length of the molecules in dilute solution. This structure appears at concentrations much lower than the overlap threshold concentration, and disappears when the concentration is sufficiently high. Liquidlike order reappears at higher concentrations on a lengthscale of the order of the thickness of the rods. The predictions of the theory for the static structure factor in tobacco mosaic virus solutions are in good agreement with light scattering experiments in both dilute and semidilute solutions. The theory predicts that k(max)similar to p(v), where k(max) is the position of the maximum in the structure factor and rho is the concentration, with v approximate to 1/3 and 1/2 in the dilute and semidilute regimes, respectively. These values are consistent with experimental results. Predictions are also presented for rigid starlike polymers.