In this article, we propose techniques for modeling and rendering of heterogeneous translucent materials that enable acquisition from measured samples, interactive editing of material attributes, and real-time rendering. The materials are assumed to be optically dense such that multiple scattering can be approximated by a diffusion process described by the diffusion equation. For modeling heterogeneous materials, we present the inverse diffusion algorithm for acquiring material properties from appearance measurements. This modeling algorithm incorporates a regularizer to handle the ill-conditioning of the inverse problem, an adjoint method to dramatically reduce the computational cost, and a hierarchical GPU implementation for further speedup. To render an object with known material properties, we present the polygrid diffusion algorithm, which solves the diffusion equation with a boundary condition defined by the given illumination environment. This rendering technique is based on representation of an object by a polygrid, a grid with regular connectivity and an irregular shape, which facilitates solution of the diffusion equation in arbitrary volumes. Because of the regular connectivity, our rendering algorithm can be implemented on the GPU for real-time performance. We demonstrate our techniques by capturing materials from physical samples and performing real-time rendering and editing with these materials.