Determining the causal structure of a set of variables is critical for both scientific inquiry and decision-making. However, this is often challenging in practice due to limited interventional data. Given that randomized experiments are usually expensive to perform, we propose a general framework and theory based on optimal Bayesian experimental design to select experiments for targeted causal discovery. That is, we assume the experimenter is interested in learning some function of the unknown graph (e.g., all descendants of a target node) subject to design constraints such as limits on the number of samples and rounds of experimentation. While it is in general computationally intractable to select an optimal experimental design strategy, we provide a tractable implementation with provable guarantees on both approximation and optimization quality based on submodularity. We evaluate the efficacy of our proposed method on both synthetic and real datasets, thereby demonstrating that our method realizes considerable performance gains over baseline strategies such as random sampling.