A cohomology class of the group GL_n(Q_p) gives rise to a characteristic class of Q_p-local systems on algebraic varieties or topological spaces. It turns out that all rational primitive cohomology classes (in degrees >1) give vanishing characteristic classes for all local systems on smooth varieties over algebraically closed fields, but form an interesting invariant for etale local systems on varieties over non-closed fields, especially when the base field is p-adic as well. This is similar to the situation with Chern-Simons characteristic classes of complex local systems, that vanish for all local systems on smooth algebraic varieties, but produce an interesting invariant for other topological spaces, such as hyperbolic 3-manifolds. I will survey what is known and not known about characteristic classes of p-adic local systems, focusing on a method for computing them via p-adic Hodge theory, that relies on associating to a Q_p-local system on an algebraic variety a vector bundle on the relative Fargues-Fontaine curve of the variety. This is based on joint works with Lue Pan and George Pappas.