  Conformal field theories play a central role in modern theoretical physics
with many applications to the understanding of phase transitions, gauge
theories and even the quantum physics of gravity, through Maldacena's
celebrated holographic duality. The key analytic tool in the field is the
so-called conformal partial wave expansion, i.e. a Fourier-like decomposition
of physical quantities into a basis of partial waves for the conformal group
SO(1,d+1). This text provides an non-technical introduction to conformal field
theory and the conformal bootstrap program with some focus on the mathematical
aspects of conformal partial wave expansions. It emphasises profound relations
with modern hypergeometry, group theory and integrable models of Gaudin and
Calogero-Sutherland type.
