Does Dwork’s rationality theorem lift to a motivic rationality theorem? In 1995, Kapranov raised this question, pointing out that the answer is yes for curves. In 2001, Valery Lunts and I used birational geometry to prove that the answer is in general no in higher dimensions, at least if one interprets “motives” to be linear combinations of varieties. Loeser and Denef conjectured that the answer should nevertheless be yes if one inverts the Lefschetz class, as one does in the theory of motivic integration. Lunts and I recently gave a new obstruction to rationality, via Galois representations, which is not killed by inverting L.