The past five years of discoveries in gravitational wave astronomy have had a profound impact on cosmology. In particular, they have triggered a host of new ideas on how to probe the fundamental nature of gravity on large scales. This work is rapidly becoming a crucial pillar in the long-standing challenge to understand dark energy.  In this talk, I’ll explain the essential phenomenology of gravitational waves outside of General Relativity. I’ll talk through some of the key tests of gravity we hope to achieve with future gravitational wave data, using standard sirens, waveforms and cross-correlation with galaxies. I’ll focus on the role the LISA experiment plays in this work, due to its capabilities for detecting gravitational waves at low frequencies, and from high redshifts.