Manin's conjecture predicts the asymptotic behavior of the number of rational points of bounded anticanonical height on Fano varieties over the rational numbers. We discuss a proof of this conjecture for smooth spherical Fano threefolds, which combines Cox rings with methods of analytic number theory. This is joint work (partly in progress) withValentin Blomer, Jörg Brüdern and Giuliano Gagliardi.