In this thesis we describe a technique for animating the behavior of viscoelastic fluids such as mucus, liquid soap, pudding, toothpaste, or clay, that exhibit a combination of both fluid and solid characteristics. The technique builds upon prior Eulerian methods for animating incompressible fluids. Our method computes viscoelastic fluid behavior by supplementing the basic Navier-Stokes equations with additional terms for elastic body forces. These terms can be readily computed on rectilinear grids using a staggered discretization scheme, and the use of an Eulerian formulation easily accommodates modeling flows that undergo large deformations with topological changes. These elastic terms require computing the material strain throughout the fluid. Because the fluid simulations do not make use of an explicit reference configuration, strain is computed by integrating strain rate and advecting the results. The transition from elastic resistance to viscous flow is controlled by von Mises' yield condition, and subsequent behavior is governed by a quasi-linear plasticity model.