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Garment representation, editing and animation are challenging topics in the area of computer vision and graphics. It remains difficult for existing garment representations to achieve smooth and plausible transitions between different shapes and topologies. In this work, we introduce, DeepCloth, a unified framework for garment representation, reconstruction, animation and editing. Our unified framework contains 3 components: First, we represent the garment geometry with a "topology-aware UV-position map", which allows for the unified description of various garments with different shapes and topologies by introducing an additional topology-aware UV-mask for the UV-position map. Second, to further enable garment reconstruction and editing, we contribute a method to embed the UV-based representations into a continuous feature space, which enables garment shape reconstruction and editing by optimization and control in the latent space, respectively. Finally, we propose a garment animation method by unifying our neural garment representation with body shape and pose, which achieves plausible garment animation results leveraging the dynamic information encoded by our shape and style representation, even under drastic garment editing operations. To conclude, with DeepCloth, we move a step forward in establishing a more flexible and general 3D garment digitization framework. Experiments demonstrate that our method can achieve state-of-the-art garment representation performance compared with previous methods.