Non-compact string backgrounds and non-rational CFT

This is an introduction to the microscopic techniques of non-rational bulk and boundary conformal field theory which are needed to describe strings moving in non-compact curved backgrounds. The latter arise, e.g. in the context of AdS/CFT-like dualities and for studies of time-dependent processes. After a general outline of the central concepts, we focus on two specific but rather prototypical models: Liouville field theory and the 2D cigar. Rather than following the historical path, the presentation attempts to be systematic and self-contained.