We address the problem of robust multiuser downlink beamforming under the assumption that the transmitter has partial covariance-based channel state information (CSI). In our approach, the uncertainty on the channel covariance matrices is assumed to be confined in an ellipsoid of given size and shape, where prior knowledge about the statistical distribution of the CSI mismatch is taken into account. The goal is to minimize the transmitted power under the worst-case quality-of-service (QoS) constraints. We extend the developed robust problem to downlink beamforming in cognitive radio (CR) networks where QoS constraints apply to the users of the secondary network (SN) and interference leaked to the primary users (PUs) is required to be below a given interference threshold. We avoid the coarse approximations used by previous solutions and obtain exact reformulations for both worst-case problems based on Lagrange duality. The resulting problems can then be approximated using semidefinite relaxation (SDR). Further, we consider a popular alternative robust approach that is based on probabilistic QoS and interference constraints and show that both approaches are generally equivalent. Computer simulations show that the proposed techniques provide substantial performance improvements over earlier robust downlink beamforming techniques for both the conventional and the CR scenarios.