Two-dimensional conformal field theory (CFT) has several sources: the search
for simple examples of quantum field theory, the description of surface
critical phenomena, the study of (super)string vacua. In the present overview
of the subject we emphasize the role of CFT in bridging the gap between
mathematics and quantum field theory and discuss some new physical concepts
that emerged in the study of CFT models: anomalous dimensions, rational CFT,
braid group statistics. In an aside, at the end of the paper, we share the
misgivings, recently expressed by Penrose, about some dominant trends in
fundamental theoretical physics.