Splitting a volumetric object is a useful operation in volume visualization and volume animation, but is not widely supported by existing systems for volume-based modeling and rendering. In this paper, we present an investigation into two main algorithmic approaches, namely explicit and implicit splitting, for modeling and rendering splitting actions in the context of both volume visualization and volume animation. We consider a generalized notion based on scalar fields, which encompasses discrete specifications (e.g., volume data sets) as well as procedural specifications (e.g., hypertextures) of volumetric objects. We examine the correctness, effectiveness, efficiency and deficiencies of each approach in specifying and controlling a spatial and temporal specification of splitting. We propose methods for implementing these approaches and for overcoming their deficiencies. We demonstrate the use of these approaches with examples of medical visualization, volume animation and special effects.