Generalized symmetries on the lattice

There exist exotic quantum zero temperature phases of matter, such as the integer and fractional quantum Hall states, which are distinguished on the basis of quantized topological invariants.  This is a well known story going back to the 1980's.  More recently, however, the same topological ideas and principles have been used to classify so-called generalized symmetries of lattice systems themselves, irrespective of any particular Hamiltonian.  I will discuss the progress in this direction, in the context of `quantum cellular automata' (QCA), dynamical Floquet phases, anomalies on the lattice, and chiral gauge theories.  The talk will be pedagogical, without much technical background required.