The Standard Model of particle physics has enjoyed unprecedented success in predicting experimental results. However, evidence from astrophysical observations points to the existence of a dark sector of particles that interact only very weakly with the Standard Model. In this work, we search for dark sector signatures in X-ray telescope data. Much of this work concerns a class of hypothetical particles, the axion-like particle (ALP). ALPs are a theoretically well-motivated extension of the Standard Model. If ALPs exist, they may lead to intriguing astrophysical signatures: in the presence of a background magnetic field, ALPs and photons can interconvert. We could detect ALPs by searching for photon to ALP conversion. For example, photons produced by point sources in or behind galaxy clusters may convert to ALPs in the cluster's magnetic field. This could lead to observable spectral anomalies. Using this strategy, we place world leading bounds on the ALP-photon coupling. One potential signal of dark matter is an anomalous line in the spectra of galaxies and galaxy clusters. In 2014, an anomalous line was found at an energy of 3.5 keV. The nature and cause of this line is still under discussion. We analyse a scenario in which the 3.5 keV line arises from dark matter decay to ALPs, which interconvert with 3.5 keV photons in astrophysical magnetic fields. We further report an anomalous deficit at 3.5 keV in the spectrum of the Active Galactic Nucleus at the centre of the Perseus galaxy cluster. This motivates the study of a new model in which both features are caused by “fluorescent dark matter” which resonantly interacts with 3.5 keV photons. We analyse observations of Perseus at 3.5 keV to date, and show that they are well explained by this model. Further theoretical and experimental work is needed to discover or exclude fundamental physics effects in X-ray spectra.