The finite element method has revolutionized structural analysis since its inception over 50 years ago, by enabling the computer analysis of geometrically complex structures. The main requirement of the finite element method is that an appropriate partition, or mesh, of the structure be created first. The elements of the partition typically have standard shapes, such as the hexahedron, pentahedron, 180and tetrahedron. While this small library of standard element shapes is sufficient for many applications, there is a growing need for more general polyhedral shapes, ones that can have an arbitrary number of vertices, edges, and faces, and ones that can be non-convex. In this chapter, we discuss current and possible future applications of polyhedral finite elements in solid mechanics. These applications include rapid engineering analysis through novel meshing and discretization techniques, and fracture and fragmentation modeling. Several finite element formulations of general polyhedra have been developed. In this chapter we use a polyhedral formulation based on the use of harmonic shape functions. Harmonic shape functions are one example of several possible generalized barycentric coordinates, as discussed in Chapter 1.