We present theoretical results to describe the crossover behavior of the viscosity of dilute and semidilute polyelectrolyte solutions. We have resolved the puzzle of the experimentally observed nonmonotonic dependence of the reduced viscosity of polyelectrolyte solutions on the polymer concentration below the entanglement limit. We use a combination of the theory of triple screening of excluded volume, electrostatic, and hydrodynamic interactions in polyelectrolyte solutions and a thermodynamic theory of interchain interactions in obtaining the crossover formulas. The calculated results are in qualitative agreement with data in the experimental literature.