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Robot navigation in crowded public spaces is a complex task that requires addressing a variety of engineering and human factors challenges. These challenges have motivated a great amount of research resulting in important developments for the fields of robotics and human-robot interaction over the past three decades. Despite the significant progress and the massive recent interest, we observe a number of significant remaining challenges that prohibit the seamless deployment of autonomous robots in crowded environments. In this survey article, we organize existing challenges into a set of categories related to broader open problems in robot planning, behavior design, and evaluation methodologies. Within these categories, we review past work and offer directions for future research. Our work builds upon and extends earlier survey efforts by (a) taking a critical perspective and diagnosing fundamental limitations of adopted practices in the field and (b) offering constructive feedback and ideas that could inspire research in the field over the coming decade.