We propose clipping methods that are capable of using complex geometries for volume clipping. The clipping tests exploit per-fragment operations on the graphics hardware to achieve high frame rates. In combination with texture-based volume rendering, these techniques enable the user to interactively select and explore regions of the data set. We present depth-based clipping techniques that analyze the depth structure of the boundary representation of the clip geometry to decide which parts of the volume have to be clipped. In another approach, a voxelized clip object is used to identify the clipped regions. Furthermore, the combination of volume clipping and volume shading is considered. An optical model is introduced to merge aspects of surface-based and volume-based illumination in order to achieve a consistent shading of the clipping surface. It is demonstrated how this model can be efficiently incorporated in the aforementioned clipping techniques.