Consider smooth projective varieties that are built geometrically using a finite number of affine spaces. Examples include projective spaces and Grassmannians, toric varieties, even-dimensional intersections of two quadrics, and moduli spaces of pointed rational curves. We consider rationality questions over non-closed fields and in the presence of group actions. Even over the real numbers, there are many open problems! (joint with Kollár, Tschinkel, and Zhang)