The nature of dark matter, which makes up more than 80% of the Universe's matter content, remains unknown. Light axions and axion-like particles (ALPs) are well motivated dark-matter candidates that could be detected through their oscillations into photons in the presence of magnetic fields. Here, complementary laboratory and astrophysical searches for dark-matter axions and ALPs are proposed that will cover more than 10 orders of magnitude of possible axion and ALP masses. The astrophysical searches will focus on high-energy gamma-ray observations with the Fermi Large Area Telescope as well as current and future imaging air Cherenkov telescopes. Photon-ALP oscillations would cause features in the spectra of distant galaxies as well as gamma-ray bursts from core-collapse supernovae. Axion and ALP decay would also increase the opacity of the Universe for gamma rays. These signals will be searched for through novel comparisons of gamma-ray data and model predictions. The laboratory searches will focus on contributions to the Any Light Particle Search (ALPS II) and International Axion Observatory (IAXO) experiments. New analysis and simulation frameworks, as well as trigger concepts, will be developed in order to significantly improve the background rejection for the Transition Edge Sensor (TES) detector employed in the ALPS experiment. These improvements could pave the way for an ALP detection in the laboratory with first data runs at the ALPS II experiment planned in 2021. Monte Carlo simulations will be used to assess whether TES detectors can achieve the low background rates required for IAXO. Such high energy resolution detectors could help to precisely measure the axion/ALP mass through mass-dependent spectral features. Through an unprecedented investigation of axion and ALP signatures and by enhancing the sensitivity of future laboratory experiments, the proposed research will discover or rule out so-far unprobed dark-matter axions and ALPs.