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We study the problem of object detection from a novel perspective in which annotation budget constraints are taken into consideration, appropriately coined Budget Aware Object Detection (BAOD). When provided with a fixed budget, we propose a strategy for building a diverse and informative dataset that can be used to optimally train a robust detector. We investigate both optimization and learning-based methods to sample which images to annotate and what type of annotation (strongly or weakly supervised) to annotate them with. We adopt a hybrid supervised learning framework to train the object detector from both these types of annotation. We conduct a comprehensive empirical study showing that a handcrafted optimization method outperforms other selection techniques including random sampling, uncertainty sampling and active learning. By combining an optimal image/annotation selection scheme with hybrid supervised learning to solve the BAOD problem, we show that one can achieve the performance of a strongly supervised detector on PASCAL-VOC 2007 while saving 12.8% of its original annotation budget. Furthermore, when $100\%$ of the budget is used, it surpasses this performance by 2.0 mAP percentage points.