In stochastic control applications, typically only an ideal model (controlled transition kernel) is assumed and the control design is based on the given model, raising the problem of performance loss due to the mismatch between the assumed model and the actual model. Toward this end, we study continuity properties of discrete-time stochastic control problems with respect to system models (i.e., controlled transition kernels) and robustness of optimal control policies designed for incorrect models applied to the true system. We study both fully observed and partially observed setups under an infinite horizon discounted expected cost criterion. We show that continuity and robustness cannot be established the under weak convergence of transition kernels in general, but that the expected induced cost is robust under total variation in that it is continuous in the mismatch of transition kernels under convergence in total variation. By imposing further assumptions on the measurement models and on the kernel itself, we show that the optimal cost can be made continuous under weak convergence of transition kernels as well. Using these continuity properties, we establish convergence results and error bounds due to mismatch that occurs by the application of a control policy which is designed for an incorrectly estimated system model to a true model, thus establishing positive and negative results on robustness. Compared to the existing literature, we obtain refined robustness results that are applicable even when the incorrect models can be investigated under weak convergence and setwise convergence criteria (with respect to a true model), in addition to the total variation criteria. These lead to practically important results on empirical learning in (datadriven) stochastic control since often, in many applications, system models are learned through training data.