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3D Garment modeling is a critical and challenging topic in the area of computer vision and graphics, with increasing attention focused on garment representation learning, garment reconstruction, and controllable garment manipulation, whereas existing methods were constrained to model garments under specific categories or with relatively simple topologies. In this paper, we propose a novel Neural Sewing Machine (NSM), a learning-based framework for structure-preserving 3D garment modeling, which is capable of learning representations for garments with diverse shapes and topologies and is successfully applied to 3D garment reconstruction and controllable manipulation. To model generic garments, we first obtain sewing pattern embedding via a unified sewing pattern encoding module, as the sewing pattern can accurately describe the intrinsic structure and the topology of the 3D garment. Then we use a 3D garment decoder to decode the sewing pattern embedding into a 3D garment using the UV-position maps with masks. To preserve the intrinsic structure of the predicted 3D garment, we introduce an inner-panel structure-preserving loss, an inter-panel structure-preserving loss, and a surface-normal loss in the learning process of our framework. We evaluate NSM on the public 3D garment dataset with sewing patterns with diverse garment shapes and categories. Extensive experiments demonstrate that the proposed NSM is capable of representing 3D garments under diverse garment shapes and topologies, realistically reconstructing 3D garments from 2D images with the preserved structure, and accurately manipulating the 3D garment categories, shapes, and topologies, outperforming the state-of-the-art methods by a clear margin.