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Deep neural network quantization with adaptive bitwidths has gained increasing attention due to the ease of model deployment on various platforms with different resource budgets. In this paper, we propose a meta-learning approach to achieve this goal. Specifically, we propose MEBQAT, a simple yet effective way of bitwidth-adaptive quantization aware training (QAT) where meta-learning is effectively combined with QAT by redefining meta-learning tasks to incorporate bitwidths. After being deployed on a platform, MEBQAT allows the (meta-)trained model to be quantized to any candidate bitwidth then helps to conduct inference without much accuracy drop from quantization. Moreover, with a few-shot learning scenario, MEBQAT can also adapt a model to any bitwidth as well as any unseen target classes by adding conventional optimization or metric-based meta-learning. We design variants of MEBQAT to support both (1) a bitwidth-adaptive quantization scenario and (2) a new few-shot learning scenario where both quantization bitwidths and target classes are jointly adapted. We experimentally demonstrate their validity in multiple QAT schemes. By comparing their performance to (bitwidth-dedicated) QAT, existing bitwidth adaptive QAT and vanilla meta-learning, we find that merging bitwidths into meta-learning tasks achieves a higher level of robustness.