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Despite the impressive performance of Artificial Intelligence (AI) systems, their robustness remains elusive and constitutes a key issue that impedes large-scale adoption. Robustness has been studied in many domains of AI, yet with different interpretations across domains and contexts. In this work, we systematically survey the recent progress to provide a reconciled terminology of concepts around AI robustness. We introduce three taxonomies to organize and describe the literature both from a fundamental and applied point of view: 1) robustness by methods and approaches in different phases of the machine learning pipeline; 2) robustness for specific model architectures, tasks, and systems; and in addition, 3) robustness assessment methodologies and insights, particularly the trade-offs with other trustworthiness properties. Finally, we identify and discuss research gaps and opportunities and give an outlook on the field. We highlight the central role of humans in evaluating and enhancing AI robustness, considering the necessary knowledge humans can provide, and discuss the need for better understanding practices and developing supportive tools in the future.