Axions and other very light axion-like particles (ALPs) appear in many extensions of the Standard Model and are well motivated theoretically: ALPs can solve the well-known strong CP problem, act as a dark matter candidate and could also explain the famous muon (g-2) discrepancy. The experimental effort to search for ALPs as dark matter candidates is ongoing and has been considerably intensified in recent years, leading to the proposal and construction of a wide range of dedicated experiments. However, none of these dedicated experiments is sensitive to those ALPs that can explain low-energy anomalies such as (g-2). I propose therefore to pioneer an alternative search strategy for axion-like particles via their decay into two photons, using data collected at the Large Hadron Collider. This approach requires fundamental innovations on the photon identification capabilities of the current detectors as well as radically new analysis strategies. Within the LightAtLHC project, I will study proton-proton and lead-lead collisions, collected during LHC Run-3, and search for Higgs Boson decays in two ALPs as well as the direct production of ALPs via photon fusion and their subsequent decay into two low-energy photons. To achieve the required sensitivity, I will develop highly specialized photon reconstruction algorithms for the ATLAS detector. These efforts will largely cover the relevant parameter space, leaving out only a small region. To also close this gap, I will extend the upcoming FASER experiment at the LHC by an innovative presampler detector, which allows for an unambiguous ALPs detection. By the end of the LightAtLHC project, I can either rule out the most promising ALP models in a mass range from 10 MeV to 1 TeV, or discover a new elementary particle.