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Axions, hypothetical low-energy particles that are often introduced to resolve open problems in particle physics and cosmology, represent one of the most well-motivated targets in searches for physics beyond the Standard Model. In this talk I will demonstrate how extreme compact objects, in particular neutron stars and black holes, naturally host environments capable of producing striking signatures of these elusive particles. These indirect searches have a strong complementarity with on-going and future laboratory experiments, and will play a key role in the hunt for axions over the coming decade.