The Jupiter Trojan asteroids are a key population for understanding the chemical and dynamical evolution of the Solar System. Surface compositions of Trojans, in turn, provide crucial information for reconstructing their histories. NASA’s Lucy mission will soon complete the first spacecraft reconnaissance of this population. This review summarizes the current state of knowledge of Trojan surface compositions and looks ahead to expected advances in that knowledge from Lucy. Surface compositions of Trojans remain uncertain due to a relative lack of diagnostic absorption features, though dedicated observations have begun to provide some clues to compositions. Trojans have uniformly low albedos, with a population average of ∼5.3%, and red spectral slopes at ultraviolet, visible, and near-infrared wavelengths. A bimodality of spectral slopes has been detected and confirmed across all these wavelengths, and the ratio of “less-red” to “red” Trojans increases with decreasing size. A broad absorption at ∼3.1 μm in some less-red Trojans may indicate the presence of N-H bearing material. Mid-infrared emissivity spectra reveal the presence of fine-grained anhydrous silicates on the surfaces. The meteorite collection contains no identifiable analogs to Trojan asteroids. Among small body populations, some Main Belt asteroids, comets, irregular satellites, and Centaurs provide reasonable spectral matches, supporting some genetic relationships among some members of these groups. The cause of the observed spectral properties remains uncertain, but recent suggestions include a combination of volatile ice sublimation and space weathering or a combination of impact gardening and space weathering. The Lucy mission will provide detailed compositional analysis of (3548) Eurybates, (15094) Polymele, (11351) Leucus, (21900) Orus, and (617) Patroclus-Menoetius, a suite of targets that sample the diversity among the Trojan population along several dimensions. With these flybys, the Lucy mission is poised to resolve many of the outstanding questions regarding Trojan surface compositions, thereby revealing how the Trojans formed and evolved and providing a clearer view of Solar System history.