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Deep image compression performs better than conventional codecs, such as JPEG, on natural images. However, deep image compression is learning-based and encounters a problem: the compression performance deteriorates significantly for out-of-domain images. In this study, we highlight this problem and address a novel task: universal deep image compression. This task aims to compress images belonging to arbitrary domains, such as natural images, line drawings, and comics. To address this problem, we propose a content-adaptive optimization framework; this framework uses a pre-trained compression model and adapts the model to a target image during compression. Adapters are inserted into the decoder of the model. For each input image, our framework optimizes the latent representation extracted by the encoder and the adapter parameters in terms of rate-distortion. The adapter parameters are additionally transmitted per image. For the experiments, a benchmark dataset containing uncompressed images of four domains (natural images, line drawings, comics, and vector arts) is constructed and the proposed universal deep compression is evaluated. Finally, the proposed model is compared with non-adaptive and existing adaptive compression models. The comparison reveals that the proposed model outperforms these. The code and dataset are publicly available at https://github.com/kktsubota/universal-dic.