Splitting a volumetric object is a useful operation in volume graphics and its applications, 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. 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 present a modeling tool for creating specifications of splitting functions, and describe the use of volume scene graphs for facilitating direct rendering of volume splitting. We demonstrate the use of these approaches with examples of volume visualization, medical illustration, volume animation, and special effects.