  We extend our earlier work on anomalies in the space of coupling constants to
four-dimensional gauge theories. Pure Yang-Mills theory (without matter) with a
simple and simply connected gauge group has a mixed anomaly between its
one-form global symmetry (associated with the center) and the periodicity of
the $\theta$-parameter. This anomaly is at the root of many recently discovered
properties of these theories, including their phase transitions and interfaces.
These new anomalies can be used to extend this understanding to systems without
discrete symmetries (such as time-reversal). We also study $SU(N)$ and $Sp(N)$
gauge theories with matter in the fundamental representation. Here we find a
mixed anomaly between the flavor symmetry group and the $\theta$-periodicity.
Again, this anomaly unifies distinct recently-discovered phenomena in these
theories and controls phase transitions and the dynamics on interfaces.
