  We study the 't Hooft anomalies of four-dimensional superconformal field
theories that arise from M5-branes wrapped on a punctured Riemann surface. In
general there are two independent contributions to the anomalies. There is a
bulk term obtained by integrating the anomaly polynomial of the world-volume
theory on the M5-branes over the Riemann surface; this contribution knows about
the punctures only through its dependence on the Euler characteristic of the
surface. The second set of contributions comes from local data at the
punctures; these terms are independent from the bulk data of the surface. Using
anomaly inflow in M-theory, we describe the general structure of the anomalies
for cases when the four-dimensional theories preserve N=2 supersymmetry. We
additionally discuss the anomalies corresponding to (p,q) punctures in N=1
theories.
