The classical Banach-Stone theorem describes the general form of all surjective linear isometries between the spaces of all continuous scalar-valued functions on compact Hausdorff spaces. It has a substantial generalization in the noncommutative setting obtained by Kadison in which he characterized the surjective linear isometries between any \( C^* \)-algebras. There are several important metrics on the positive cones in matrix algebras or in the far more general setting of \( C^* \)-algebras, the Thompson metric and the Bures metric, to name only two. In this talk, we describe the corresponding surjective isometries (not assuming any sort of linearity). If time permits, we will speak of some other kinds of symmetries of positive cones, e.g., the ones related to relative entropies.