  A $q$-form global symmetry is a global symmetry for which the charged
operators are of space-time dimension $q$; e.g. Wilson lines, surface defects,
etc., and the charged excitations have $q$ spatial dimensions; e.g. strings,
membranes, etc. Many of the properties of ordinary global symmetries ($q$=0)
apply here. They lead to Ward identities and hence to selection rules on
amplitudes. Such global symmetries can be coupled to classical background
fields and they can be gauged by summing over these classical fields. These
generalized global symmetries can be spontaneously broken (either completely or
to a subgroup). They can also have 't Hooft anomalies, which prevent us from
gauging them, but lead to 't Hooft anomaly matching conditions. Such anomalies
can also lead to anomaly inflow on various defects and exotic Symmetry
Protected Topological phases. Our analysis of these symmetries gives a new
unified perspective of many known phenomena and uncovers new results.
