In contrast to fully supervised methods using pixel-wise mask labels, box-supervised instance segmentation takes advantage of simple box annotations, which has recently attracted increasing research attention. This paper presents a novel single-shot instance segmentation approach, namely Box2Mask, which integrates the classical level-set evolution model into deep neural network learning to achieve accurate mask prediction with only bounding box supervision. Specifically, both the input image and its deep features are employed to evolve the level-set curves implicitly, and a local consistency module based on a pixel affinity kernel is used to mine the local context and spatial relations. Two types of single-stage frameworks, i.e., CNN-based and transformer-based frameworks, are developed to empower the level-set evolution for box-supervised instance segmentation, and each framework consists of three essential components: instance-aware decoder, box-level matching assignment and level-set evolution. By minimizing the level-set energy function, the mask map of each instance can be iteratively optimized within its bounding box annotation. The experimental results on five challenging testbeds, covering general scenes, remote sensing, medical and scene text images, demonstrate the outstanding performance of our proposed Box2Mask approach for box-supervised instance segmentation. In particular, with the Swin-Transformer large backbone, our Box2Mask obtains 42.4% mask AP on COCO, which is on par with the recently developed fully mask-supervised methods.