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In this paper we present an end-to-end meta-learned system for image compression. Traditional machine learning based approaches to image compression train one or more neural network for generalization performance. However, at inference time, the encoder or the latent tensor output by the encoder can be optimized for each test image. This optimization can be regarded as a form of adaptation or benevolent overfitting to the input content. In order to reduce the gap between training and inference conditions, we propose a new training paradigm for learned image compression, which is based on meta-learning. In a first phase, the neural networks are trained normally. In a second phase, the Model-Agnostic Meta-learning approach is adapted to the specific case of image compression, where the inner-loop performs latent tensor overfitting, and the outer loop updates both encoder and decoder neural networks based on the overfitting performance. Furthermore, after meta-learning, we propose to overfit and cluster the bias terms of the decoder on training image patches, so that at inference time the optimal content-specific bias terms can be selected at encoder-side. Finally, we propose a new probability model for lossless compression, which combines concepts from both multi-scale and super-resolution probability model approaches. We show the benefits of all our proposed ideas via carefully designed experiments.