  Starting with a gentle approach to the AGT correspondence from its 6d origin,
these notes provide a wide (albeit shallow) survey of the literature on
numerous extensions of the correspondence up to early 2020. This is an extended
writeup of the lectures given at the Winter School YRISW 2020, to appear in a
special issue of JPhysA.
  Class S is a wide class of 4d $N=2$ supersymmetric gauge theories (ranging
from super-QCD to non-Lagrangian theories) obtained by twisted compactification
of 6d $N=(2,0)$ superconformal theories on a Riemann surface. This 6d
construction yields the Coulomb branch and Seiberg-Witten geometry of class S
theories, geometrizes S-duality, and leads to the AGT correspondence, which
states that many observables of class S theories are equal to 2d CFT
correlators. For instance, the four-sphere partition function of a 4d $N=2$
SU(2) superconformal quiver theory is equal to a Liouville CFT correlator of
primary operators.
  Extensions of the AGT correspondence abound: asymptotically-free gauge
theories and Argyres-Douglas theories correspond to irregular CFT operators,
quivers with higher-rank gauge groups and non-Lagrangian tinkertoys such as
$T_N$ correspond to Toda CFT correlators, and nonlocal operators (Wilson-'t
Hooft loops, surface operators, domain walls) correspond to Verlinde networks,
degenerate primary operators, braiding and fusion kernels, and Riemann surfaces
with boundaries.
