We explore the higher-form generalized global symmetries
and higher anomalies based on a generalized cobordism theory.
Our cobordism calculation guides us to classify
higher anomalies and topological terms for
Yang-Mills (YM) gauge theories
and sigma models in various dimensions.
Some of YM gauge theories can be obtained from dynamically gauging the SU(N) time-reversal symmetric cobordism invariants
(SU(N)-generalized topological superconductors/insulators [arXiv:1711.11587]). We elaborate the cases of YM theory with a compact Lie gauge group (such as SU(N) with N=2, 3 and others) particularly in a 4d spacetime.
We provide the relevant homotopy and cobordism group calculations of higher classifying spaces,
based on mathematical tools of algebraic topology,
to support the physics stories.