Spectroscopic observations from the ultraviolet to the mid-infrared have revealed new and diagnostic differences among primitive asteroids. We review the spectral characteristics of these asteroids and their inferred compositional and physical properties. Primitive asteroids throughout the belt show carbon-rich compounds, varying degrees of aqueous alteration and even surface ice; recent observations provide significant new constraints on composition, thermal inertia, and other surface properties. New mid-infrared connections between primitive asteroids and interplanetary dust particles indicate that the latter sample a larger fraction of main belt asteroids than meteorites. Links with the composition of comets are consistent with a proposed continuum between primitive asteroids and comets. Two sample-return missions, OSIRIS-REx and Hayabusa 2, will visit primitive near-Earth asteroids (NEAs). Most spacecraft-accessible NEAs originate in the inner asteroid belt, which contains several primitive asteroid families and a background of primitive asteroids outside these families. Initial results from these families offer a tantalizing preview of the properties expected in the NEAs they produce. So far, primitive asteroids in the inner belt fall into two spectral groups. The first group includes the Polana-Eulalia families, which show considerable spectral homogeneity in spite of their dynamical and collisional complexity. In contrast, the Erigone and Sulamitis families are spectrally diverse and most of their members show clear 0.7 microns hydration features. The two sample-return targets (101955) Bennu and (162173) Ryugu, most likely originated in the Polana family.