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These are lecture notes of a course given at the Summer School on Topology and Field Theories held at the Centre for Mathematics of the University of Notre Dame, Indiana, from May 29 to June 2, 2012. The idea of extending quantum field theories to manifolds of lower dimension was first proposed by Dan Freed in the nineties. In the case of conformal field theory (CFT), we are talking of an extension of the Atiyah-Segal axioms, where one replaces the bordism category of Riemann surfaces by aarXiv:1304.7328v2 fatcat:pfthcwliurh2vbdojc7ppsx7pi