A sampled object representation (SOR) defines a graphical model using data obtained from a sampling process, which takes a collection of samples at discrete positions in space in order to capture certain geometrical and physical properties of one or more objects of interest. Examples of SORs include images, videos, volume datasets and point datasets. Unlike many commonly used data representations in computer graphics, SORs lack in geometrical, topological and semantic information, which is much needed for controlling deformation and animation. Hence it poses a significant scientific and technical challenge to develop deformation and animation methods that operate upon SORs. Such methods can enable computer graphics and computer animation to benefit enormously from the advances of digital imaging technology. In this state of the art report, we survey a wide range of techniques that have been developed for manipulating, deforming and animating SORs. We consider a collection of elementary operations for manipulating SORs, which can serve as building blocks of deformation and animation techniques. We examine a collection of techniques that are designed to transform the geometry shape of deformable objects in sampled representations and pay particular attention to their deployment in surgical simulation. We review a collection of techniques for animating digital characters in SORs, focusing on recent developments in volume animation.