Title: Learn Appropriate Precise Distributions for Binary Neural Networks. Abstract: Binary Neural Networks (BNNs) have shown great promise for real-world embedded devices. However, BNNs always suffer from obtaining unsatisfactory accuracy performance on a large dataset such as ImageNet, which could hinder their further widespread applications in practice. Nevertheless, enhancing BNN's performance is extremely challenging owing to its limited capacity. Several distillation approaches in which the knowledge of a real-valued teacher model is distilled to a binary student network have been proposed to boost one BNN's accuracy. However, directly employing previous distillation solutions yields inferior results due to an unsuitable match between the representational capacity of the adopted real-valued teacher model and target binary student network. In this work, we reexamine the design of knowledge distillation framework specially for BNNs and test the limits of what a pure BNN can achieve. We firstly define one group which consists of multi real-valued networks owning particular properties, and then introduce a distribution-specific loss to enforce the binary network to mimic the distribution of one real-valued network fetched from this group in a certain order. In addition, we propose one distance-aware combinational model to provide one binary network with more comprehensive guidance, and present related suitable training strategies. The BNN in this built knowledge distillation framework can be facilitated to learn appropriate precise distributions, dubbed APD-BNN. As a result, APD-BNN can reach its performance limit while incurring no additional computational cost. Compared with the state-of-the-art BNNs, APD-BNN can obtain up to 1.4$\%$ higher accuracy on the ImageNet dataset with using the same architecture. Specifically, APD-BNN is capable of gaining 72.0$\%$ top-1 accuracy on ImageNet with only 87M OPs. Thus, it achieves the same accuracy of existing official real-valued MobileNetV2 at 71$\%$ fewer OPs, demonstrating the huge potential to apply BNNs in practice. Our code and models will be available.