So far (September 1, 2021) more than 50 gravitational waves (GWs), emitted by binary systems of black hole - black hole (BBH), or black hole - neutron star (BBN), or neutron star - neutron star (BNS), have been observed by LIGO and Virgo collaborations, since the first detection in the human history by LIGO on September 14, 2015. The 2017 physics Nobel Prize was awarded to three leaders of the observation, Profs. Rainer Weiss, Barry
Barish and Kip Thorne, while half of the 2020 physics Nobel Prize went to Prof. Roger Penrose ``for the discovery that black hole formation is a robust prediction of the general theory of relativity." Soon, observations of GWs will become routine, and their applications to
understanding fundamental physics just start. One of such applications is to test gravitational theories by GW observations.

In this talk, I shall present our recent studies on testing Einstein-aether theory by GWs. Einstein-aether theory is a particular type of the general vector-tensor theory, in which the vector field, the so-called aether, is timelike and unity, so it defines a preferred direction at each given moment and spatial point. As a result, it locally breaks the Lorentz symmetry - a cornerstone of modern physics. Yet, it is self-consistent (free of ghosts, instabilities, and so on), and is consistent with all the observations and experiments. Its Cauchy problem is also well imposed. In addition, due to the presence of the aether, three different species of gravitons exist, the spin-0, spin-1, and spin-2 gravitons, while, in Einstein's general theory of relativity (GR), only spin-2 gravitons exist. This will bring dramatical differences
especially in the strong-field regime.

In this talk, I shall first provide a basic overview on GWs in GR, and then report what we have found in Einstein-aether theory. In particular, I shall show that the GW observations already reduce the four free coupling constants of the theory to two, and future studies of GWs emitted in the inspiral and ringdown phases shall further reduce the parameter space of the theory.