Title: Once Quantized for All: Progressively Searching for Quantized Compact Models. Abstract: Automatic search of Quantized Neural Networks (QNN) has attracted a lot of attention. However, the existing quantization-aware Neural Architecture Search (NAS) approaches inherit a two-stage search-retrain schema, which is not only time-consuming but also adversely affected by the unreliable ranking of architectures during the search. To avoid the undesirable effect of the search-retrain schema, we present Once Quantized for All (OQA), a novel framework that searches for quantized compact models and deploys their quantized weights at the same time without additional post-process. While supporting a huge architecture search space, our OQA can produce a series of quantized compact models under ultra-low bit-widths(e.g. 4/3/2 bit). A progressive bit inheritance procedure is introduced to support ultra-low bit-width. Our searched model family, OQANets, achieves a new state-of-the-art (SOTA) on quantized compact models compared with various quantization methods and bit-widths. In particular, OQA2bit-L achieves 64.0\% ImageNet Top-1 accuracy, outperforming its 2 bit counterpart EfficientNet-B0@QKD by a large margin of 14\%  using 30\% less computation cost. 