We consider the downlink of a cellular system in which the base station employs multiple transmit antennas, each receiver has a single antenna, and the users specify. We consider communication schemes in which the users have certain Quality of Service (QoS) requirements. We study the design of robust broadcasting schemes that minimize the transmission power necessary to guarantee that the QoS requirements are satisfied for all channels within bounded uncertainty regions around the transmitter's estimate of each user's channel. Each user's QoS requirement is formulated as a constraint on the mean square error (MSE) in its received signal, and we show that these MSE constraints imply constraints on the received SINR. Using the MSE constraints, we present a unified design approach for robust linear and non-linear transceivers with QoS requirements. The proposed designs overcome the limitations of existing approaches that provide conservative designs or are only applicable to the case of linear precoding. Furthermore, we provide computationally-efficient design formulations for a rather general model of channel uncertainty that subsumes many natural choices for the uncertainty region. We also consider the problem of the robust counterpart to precoding schemes that maximize the fidelity of the weakest user's signal subject to a power constraint. For this problem, we provide quasi-convex formulations, for both linear and non-linear transceivers, that can be efficiently solved using a one-dimensional bisection search. Our numerical results demonstrate that in the presence of CSI uncertainty, the proposed designs provide guarantees for a larger range of QoS requirements than the existing approaches, and consume require less transmission power in providing these guarantees.