The notion of G-varieties was introduced by Manin when he studied rationality problems of surfaces over perfect fields in the 1960s. A G-variety is an algebraic variety carrying a generically free regular G-action. There are close connections, as well as drastic differences between birational geometry of G-varieties and that of varieties over non-closed fields. In this talk, I will explain their similarities and differences through our work on equivariant birational geometry of Fano threefolds of large index (e.g., quadrics, cubics, intersections of two quadrics), with an emphasis on their equivariant unirationalities.  This is joint work with Yuri Tschinkel and Ivan Cheltsov.