In this talk, I will present recent work on high-frequency solutions to the Einstein vacuum equations. From a physical point of view, these solutions model high-frequency gravitational waves and describe how waves travel on a fixed background metric. There are also interested when studying the Burnett conjecture, which adresses the lack of compactness of the family of vacuum spacetimes. These high-frequency spacetimes are singular and require to work under the regime of well-posedness for the Einstein vacuum equations. I will review the literature on the subject and then show how one can construct them in generalised wave gauge by defining high-frequency ansatz.