Built between 1965 and '76 when Brutalist architecture was all the rage, it was designed by Chamberlin, Powell and Bon and constructed to serve as council residences after the World War II bombings. These guys thought of everything and ingeniously incorporated multi-use complex including the Barbican Arts Centre, the Barbican Library (which I haven't found yet) a conservatory and some amazing outdoor spaces. Today it's the home to some of London's biggest ballers, and has a cafe that serves some decent sandwiches.
A project on the Barbican I'm jealous of. Worth checking out.