In this paper, we propose a novel predictive model, active volume model (AVM), for object boundary extraction. It is a dynamic "object" model whose manifestation includes a deformable curve or surface representing a shape, a volumetric interior carrying appearance statistics, and an embedded classifier that separates object from background based on current feature information. The model focuses on an accurate representation of the foreground object's attributes, and does not explicitly represent the background. As we will show, however, the model is capable of reasoning about the background statistics thus can detect when is change sufficient to invoke a boundary decision. When applied to object segmentation, the model alternates between two basic operations: 1) deforming according to current region of interest (ROI), which is a binary mask representing the object region predicted by the current model, and 2) predicting ROI according to current appearance statistics of the model. To further improve robustness and accuracy when segmenting multiple objects or an object with multiple parts, we also propose multiple-surface active volume model (MSAVM), which consists of several single-surface AVM models subject to high-level geometric spatial constraints. An AVM's deformation is derived from a linear system based on finite element method (FEM). To keep the model's surface triangulation optimized, surface remeshing is derived from another linear system based on Laplacian mesh optimization (LMO). Thus efficient optimization and fast convergence of the model are achieved by solving two linear systems. Segmentation, validation and comparison results are presented from experiments on a variety of 2-D and 3-D medical images.