Context. Dynamical and albedo properties suggest that asteroids in cometary orbits (ACOs) are dormant or extinct comets. Their study provides new insights for understanding the end-states of comets and the size of the comet population. Aims: We intend to study the visible and near-infrared (NIR) spectral properties of different ACO populations and compare them to the independently determined properties of comets. Methods: We select our ACOs sample based on published dynamical criteria and present our own observational results obtained using the 10.4 m Gran Telescopio Canarias (GTC), the 4.2 m William Herschel Telescope (WHT), the 3.56 m Telescopio Nazionale Galileo (TNG), and the 2.5 m Isaac Newton Telescope (INT), all located at the El Roque de los Muchachos Observatory (La Palma, Spain), and the 3.0 m NASA Infrared Telescope Facility (IRTF), located at the Mauna Kea Observatory, in Hawaii. We include in the analysis the spectra of ACOs obtained from the literature. We derive the spectral class and the visible and NIR spectral slopes. We also study the presence of hydrated minerals by studying the 0.7 μm band and the UV-drop below 0.5 μm associated with phyllosilicates. Results: We present new observations of 17 ACOs, 11 of them observed in the visible, 2 in the NIR and 4 in the visible and NIR. We also discuss the spectra of 12 ACOs obtained from the literature. All but two ACOs have a primitive-like class spectrum (X or D-type). Almost 100% of the ACOs in long-period cometary orbits (Damocloids) are D-types. Those in Jupiter family comet orbits (JFC-ACOs) are ˜60% D-types and ˜40% X-types. The mean spectral slope S' of JFC-ACOs is 9.7 ± 4.6%/1000 Å and for the Damocloids this is 12.2 ± 2.0%/1000 Å. No evidence of hydration on the surface of ACOs is found from their visible spectra. The spectral slope and spectral class distribution of ACOs is similar to that of comets. Conclusions: The spectral taxonomical classification and the spectral slope distribution of ACOs, and the lack of spectral features indicative of the presence of hydrated minerals on their surface, strongly suggest that ACOs are likely dormant or extinct comets.