After a short review on the current status of gravitational wave observations, I will consider the interesting applications of gravitational wave astronomy to cosmology, focusing in particular on the possible outcomes of the eLISA mission. I will briefly discuss what kind of high-energy physics will be possible to probe with eLISA at early-times. I will then explain the concept of standard sirens and how these can be employed to probe the late-time expansion of the universe. Finally, considering realistically simulated astrophysical data, I will present eLISA forecast constraints on both standard and alternative cosmological models and compare them with current observations.