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While recent work has shown progress on extracting clothed 3D human avatars from a single image, video, or a set of 3D scans, several limitations remain. Most methods use a holistic representation to jointly model the body and clothing, which means that the clothing and body cannot be separated for applications like virtual try-on. Other methods separately model the body and clothing, but they require training from a large set of 3D clothed human meshes obtained from 3D/4D scanners or physics simulations. Our insight is that the body and clothing have different modeling requirements. While the body is well represented by a mesh-based parametric 3D model, implicit representations and neural radiance fields are better suited to capturing the large variety in shape and appearance present in clothing. Building on this insight, we propose SCARF (Segmented Clothed Avatar Radiance Field), a hybrid model combining a mesh-based body with a neural radiance field. Integrating the mesh into the volumetric rendering in combination with a differentiable rasterizer enables us to optimize SCARF directly from monocular videos, without any 3D supervision. The hybrid modeling enables SCARF to (i) animate the clothed body avatar by changing body poses (including hand articulation and facial expressions), (ii) synthesize novel views of the avatar, and (iii) transfer clothing between avatars in virtual try-on applications. We demonstrate that SCARF reconstructs clothing with higher visual quality than existing methods, that the clothing deforms with changing body pose and body shape, and that clothing can be successfully transferred between avatars of different subjects. The code and models are available at https://github.com/YadiraF/SCARF.