We will argue that the cigar conformal field theory SL(2,R)/U(1) is a prime example of a non-rational conformal field theory and we exhibit some of its characteristic features that might generalize to other non-rational conformal field theories. We discuss how this particular non-rational conformal field theory (with or without boundary) arises in string theory backgrounds with NS5-branes, and how the technical advances reached in the conformal field theory can illuminate aspects of holography in these backgrounds.