  We determine the $d+1$ dimensional topological field theory, which encodes
the higher-form symmetries and their 't Hooft anomalies for $d$-dimensional
QFTs obtained by compactifying M-theory on a non-compact space $X$. The
resulting theory, which we call the Symmetry TFT, or SymTFT for short, is
derived by reducing the topological sector of 11d supergravity on the boundary
$\partial X$ of the space $X$. Central to this endeavour is a reformulation of
supergravity in terms of differential cohomology, which allows the inclusion of
torsion in cohomology of the space $\partial X$, which in turn gives rise to
the background fields for discrete (in particular higher-form) symmetries. We
apply this framework to 7d super-Yang Mills where $X=
\mathbb{C}^2/\Gamma_{ADE}$, as well as the Sasaki-Einstein links of Calabi-Yau
three-fold cones that give rise to 5d superconformal field theories. This
M-theory analysis is complemented with a IIB 5-brane web approach, where we
derive the SymTFTs from the asymptotics of the 5-brane webs. Our methods apply
to both Lagrangian and non-Lagrangian theories, and allow for many
generalisations.
