This work studies consensus networks over finite fields, where agents process and communicate values from the set of integers {0,..., p-1}, for some prime number p, and operations are performed modulo p. For consensus networks over finite fields we provide necessary and sufficient conditions on the network topology and weights to ensure convergence. For instance we show that, differently from the case of consensus networks over the field of real numbers, consensus networks over finite fields converge in finite time, and that properties of the agents interaction graph are not sufficient to ensure finite-field consensus. Finally, we discuss the application of finite-field consensus to distributed averaging in sensor networks.