For a finite map between curves over an algebraically closed field of characteristic p, it is well understood how to measure wild ramification (Swan conductor) and use it to explain the shortfall in the Riemann-Hurwitz formula (Grothendieck-Ogg-Shafarevich formula). However, already for maps between surfaces this becomes somewhat mysterious. I'll describe a construction of a "differential Swan conductor" using ideas from the theory of crystalline/rigid cohomology. Although we cannot yet provide a global cohomological interpretation, we can at least prove integrality of the conductor (Hasse-Arf property), reconcile with an alternate definition due to Abbes-Saito, and obtain some variational properties (e.g., comparing the conductors along two transverse divisors with the conductor along the exceptional divisor of a blowup). Much of this is joint work with my student Liang Xiao.