  By developing a generalized cobordism theory, we explore the higher global
symmetries and higher anomalies of quantum field theories and interacting
fermionic/bosonic systems in condensed matter. Our essential math input is a
generalization of Thom-Madsen-Tillmann spectra, Adams spectral sequence, and
Freed-Hopkins's theorem, to incorporate higher-groups and higher classifying
spaces. We provide many examples of bordism groups with a generic $H$-structure
manifold with a higher-group $\mathbb{G}$, and their bordism invariants ---
e.g. perturbative anomalies of chiral fermions [originated from
Adler-Bell-Jackiw] or bosons with U(1) symmetry in any even spacetime
dimensions; non-perturbative global anomalies such as Witten anomaly and the
new SU(2) anomaly in 4d and 5d. Suitable $H$ such as SO/Spin/O/Pin$^\pm$
enables the study of quantum vacua of general bosonic or fermionic systems with
time-reversal or reflection symmetry on (un)orientable spacetime. Higher 't
Hooft anomalies of $d$d live on the boundary of $(d+1)$d
higher-Symmetry-Protected Topological states (SPTs) or symmetric invertible
topological orders (i.e., invertible topological quantum field theories at low
energy); thus our cobordism theory also classifies and characterizes
higher-SPTs. Examples of higher-SPT's anomalous boundary theories include
strongly coupled non-Abelian Yang-Mills gauge theories and sigma models,
complementary to physics obtained in [arXiv:1810.00844, 1812.11955, 1812.11968,
1904.00994].
