The authors present a new approach for localizing electromagnetic sources using sensors where the output of each is a vector consisting of the complete six electric and magnetic field components. Two types of source transmissions are considered: (1) single signal transmission (SST), and (2) dual signal transmission (DST). The model is given in terms of several parameters, including the wave direction of arrival (DOA) and state of polarization. A compact expression is derived for the Cramer-Rao bound (CRB) on the estimation errors of these parameters for the multi-source multi-vector-sensor model. Quality measures including mean-square angular error (MSAE) and covariance of vector angular error (CVAE) are introduced, and their lower bounds are derived. The advantage of using vector sensors is highlighted by explicit evaluation of the MSAE and CVAE bounds for source localization with a single vector sensor. A simple algorithm for estimating the source DOA with this sensor is presented along with its statistical performance analysis. >