\indent Whereas many physical systems can be identified as Hamiltonian dynamical systems, mechanical systems under rolling and sliding constraints, even simple ones such as a rolling penny, skateboard, and sleigh, are non-Hamiltonian. This is due to the fact that those constraints are nonholonomic (non-integrable). Nonholonomic constraints destroy some nice features of Hamiltonian dynamical systems, most importantly symplecticity, while retaining some Hamiltonian properties, such as energy conservation. Many concepts and ideas in Hamiltonian dynamics have been generalized from the differential-geometric point of view to incorporate nonholonomic constraints and also to explain the "almost Hamiltonian" behavior of nonholonomic systems. In this talk, I will show how to generalize Hamilton--Jacobi theory to nonholonomic systems and its application to exactly integrating the equations of motion, touching on the basic geometric concepts of nonholonomic and Hamiltonian systems and also the tools and techniques used to reconcile the differences between them.