  We consider topological defect lines (TDLs) in two-dimensional conformal
field theories. Generalizing and encompassing both global symmetries and
Verlinde lines, TDLs together with their attached defect operators provide
models of fusion categories without braiding. We study the crossing relations
of TDLs, discuss their relation to the 't Hooft anomaly, and use them to
constrain renormalization group flows to either conformal critical points or
topological quantum field theories (TQFTs). We show that if certain
non-invertible TDLs are preserved along a RG flow, then the vacuum cannot be a
non-degenerate gapped state. For various massive flows, we determine the
infrared TQFTs completely from the consideration of TDLs together with modular
invariance.
