Vibrating wire viscometers are disclosed. An example apparatus to determine the viscosity of a downhole fluid is described, the apparatus including a wire to be immersed in a downhole fluid, to vibrate when an alternating current is applied to the wire within a magnetic field, and to generate an electromotive force when vibrating within the magnetic field, the wire comprising a first resistance. The apparatus further includes a nulling circuit coupled to the wire, wherein the nulling circuit comprises a second resistance that is selectable to be substantially equal to the first resistance, and an analyzer coupled to the wire and the nulling circuit to determine the first resistance, the second resistance, and a viscosity of the downhole fluid based on the first and second resistances, at least one characteristic of the wire, and the electromotive force.