  We study generalized symmetries of quantum field theories in 1+1D generated
by topological defect lines with no inverse. This paper follows our companion
paper on gapped phases and anomalies associated with these symmetries. In the
present work we focus on identifying fusion category symmetries, using both
specialized 1+1D methods such as the modular bootstrap and (rational) conformal
field theory (CFT), as well as general methods based on gauging finite
symmetries, that extend to all dimensions. We apply these methods to $c = 1$
CFTs and uncover a rich structure. We find that even those $c = 1$ CFTs with
only finite group-like symmetries can have continuous fusion category
symmetries, and prove a Noether theorem that relates such symmetries in general
to non-local conserved currents. We also use these symmetries to derive new
constraints on RG flows between 1+1D CFTs.

High Energy Physics - Theory;Statistical Mechanics;Strongly Correlated Electrons;Quantum Algebra