Skin cancer is a major public health problem that could benefit from computer-aided diagnosis to reduce the burden of this common disease. Skin lesion segmentation from images is an important step toward achieving this goal. However, the presence of natural and artificial artifacts (e.g., hair and air bubbles), intrinsic factors (e.g., lesion shape and contrast), and variations in image acquisition conditions make skin lesion segmentation a challenging task. Recently, various researchers have explored the applicability of deep learning models to skin lesion segmentation. In this survey, we cross-examine 177 research papers that deal with deep learning-based segmentation of skin lesions. We analyze these works along several dimensions, including input data (datasets, preprocessing, and synthetic data generation), model design (architecture, modules, and losses), and evaluation aspects (data annotation requirements and segmentation performance). We discuss these dimensions both from the viewpoint of select seminal works, and from a systematic viewpoint, examining how those choices have influenced current trends, and how their limitations should be addressed. To facilitate comparisons, we summarize all examined works in a comprehensive table as well as an interactive table available online.