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The gravitational waves from merging binary systems carry unique information about the internal structure of compact objects. This is of key interest for neutron stars, whose material is compressed by their tremendous gravity to supra-nuclear densities. Under such extreme conditions we expect novel, unexplored phases of matter to emerge. In this talk, I will describe examples of how the matter properties and their interplay with strong-field gravity leads to characteristic signatures in the gravitational waves. I will also highlight new insights gained from recent gravitational-wave discoveries, and conclude with an outlook onto the remaining challenges and exciting prospects for the next years, as gravitational-wave science continues to move towards an era of precision physics.