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Deep learning based methods have significantly boosted the study of automatic building extraction from remote sensing images. However, delineating vectorized and regular building contours like a human does remains very challenging, due to the difficulty of the methodology, the diversity of building structures, and the imperfect imaging conditions. In this paper, we propose the first end-to-end learnable building contour extraction framework, named BuildMapper, which can directly and efficiently delineate building polygons just as a human does. BuildMapper consists of two main components: 1) a contour initialization module that generates initial building contours; and 2) a contour evolution module that performs both contour vertex deformation and reduction, which removes the need for complex empirical post-processing used in existing methods. In both components, we provide new ideas, including a learnable contour initialization method to replace the empirical methods, dynamic predicted and ground truth vertex pairing for the static vertex correspondence problem, and a lightweight encoder for vertex information extraction and aggregation, which benefit a general contour-based method; and a well-designed vertex classification head for building corner vertices detection, which casts light on direct structured building contour extraction. We also built a suitable large-scale building dataset, the WHU-Mix (vector) building dataset, to benefit the study of contour-based building extraction methods. The extensive experiments conducted on the WHU-Mix (vector) dataset, the WHU dataset, and the CrowdAI dataset verified that BuildMapper can achieve a state-of-the-art performance, with a higher mask average precision (AP) and boundary AP than both segmentation-based and contour-based methods.