The Spitzer Space Telescope (SST), aloft for over two years at time of writing, has so far devoted almost 600 hours of observing time to Solar System science, and small bodies make up a significant fraction of the objects that have been observed. For the first time we now have high accuracy mid-infrared data to study the fundamental mineralogical and physical properties of a large number of objects of different types. In this paper we review some of the exciting recent results derived from SST photometry (in six bands from 3.6 to 70 $\mu$m) and spectroscopy (from 5 to 40 $\mu$m) of asteroids and comets. The observations reveal their spectral energy distributions (SEDs), and we discuss three important science goals that can be addressed with these data: (1) finding compositional diagnostics of these objects, (2) determining their bulk thermal properties, and (3) understanding the surface evolution of primitive and icy bodies. We focus primarily on comet-asteroid transition objects, low-albedo asteroids, cometary nuclei, Trojans, Centaurs, and trans-Neptunian objects. In particular, we will show: emissivity features in the SEDs and identification of the compositional sources, the constraints on thermal inertia and infrared beaming through the samples of the thermal continuum, and an intercomparison of albedos across dynamically-related bodies.