A century of research on cosmic rays has revealed many of the fundamental properties of these energetic particles, such as their chemical composition and energy spectrum. The sources of cosmic rays, however, remain unknown. The discovery of a flux of high-energy astrophysical neutrinos by the IceCube observatory in 2013 may hold the key to solving this enduring question. High-energy neutrinos, as well as gamma rays, are produced in hadronic cosmic-ray interactions occurring at their source or during propagation. While no neutrino source has been identified so far, the sensitivity of these studies can be increased by searching for gamma-ray counterparts in temporal and spatial correlation with the IceCube neutrino events. The detection of gamma-ray emission in coincidence with an astrophysical neutrino would represent a smoking-gun signature of a cosmic-ray production site and potentially reveal details about the acceleration medium. In this talk I will present the status and recent results from the neutrino follow-up program of the VERITAS observatory, an air-Cherenkov telescope array in southern Arizona USA dedicated to gamma-ray astronomy in the very-high-energy range (VHE, E > 100 GeV). I will also describe prospects for neutrino follow-up observations using the Cherenkov Telescope Array, a next-generation VHE gamma-ray instrument.