It is widely accepted that drug discovery often requires a systems-level polypharmacology approach to tackle problems such as lack of efficacy and emerging resistance of single-targeted compounds. Network pharmacology approaches are increasingly being developed and applied to find new therapeutic opportunities and to re-purpose approved drugs. However, these recent advances have been relatively slow to be translated into the field of natural products. Here, we argue that a network pharmacology approach would enable an effective mapping of the yet unexplored target space of natural products, hence providing a systematic means to extend the druggable space of proteins implicated in various complex diseases. We give an overview of the key network pharmacology concepts and recent experimental-computational approaches that have been successfully applied to natural product research, including unbiased elucidation of mechanisms of action as well as systematic prediction of effective therapeutic combinations. We focus specifically on anticancer applications that use in vivo and in vitro functional phenotypic measurements, such as genome-wide transcriptomic response profiles, which enable a global modelling of the multi-target activity at the level of the biological pathways and interaction networks. We also provide representative examples of other disease applications, databases and tools as well as existing and emerging resources, which may prove useful for future natural product research. Finally, we offer our personal view of the current limitations, prospective developments and open questions in this exciting field.