Quantum Field Theories (QFTs) have a universal protected sector provided by topological operators.
They generalize the ordinary notion of symmetry in various directions, allowing to transport many
familiar concepts (symmetry breaking, anomalies, gauging, etc...) in a much more general framework.
This thesis is devoted to the study of several aspects of these topological properties, and their interplay
with the dynamics. A central tool is provided by a Topological Quantum Field Theory (TQFT) living
in one higher dimension, that encodes all the properties of the topological sector in an elegant way,
and allows to extract, from topology, dynamical constraints that would be inaccessible otherwise.
Some of the applications that we will find include the holographic dual of the so-called categorical
symmetries, constraints on the infrared imposed by generalized symmetries, the discovery of new
topological properties of certain gapless phases, a new class of exotic TQFTs, and the derivation of
the holographic dual of any Goldstone theory describing spontaneous symmetry breaking.
