N=4 supersymmetric Yang-Mills theory can be thought of as the most symmetric field theory that does not include gravity. It is one of the simplest theories and is one of the few finite quantum field theories in 4 dimensions. N=4 supersymmetric Yang-Mills theory describes a universe with both boson and fermion fields in 4 dimensions, related by 4 supersymmetries.
Montonen-Olive duality
The first kind of S-duality relationship involved this very N=4 supersymmetric Yang-Mills theory. S-duality involves a large coupling constant and a weak coupling constant. The coupling constant determines the strength of a physical theory. For example, in string theory, the coupling constant is determined by the probability of strings splitting and reconnecting. Claus Montonen and David I. Olive, proposed that the N=4 supersymmetric Yang-Mills theory with a coupling constant g, is equivalent to the same N=4 supersymmetric Yang-Mills theory with the coupling constant 1/g. This is the oldest example of this kind of a Strong-weak duality.
AdS/CFT
In the AdS/CFT correspondence, in it's initial proposal by Juan Maldacena in 1997, there is a correspondence between a bulk theory of quantum gravity with a boundary conformal field theory. The initial proposal was that the N=4 supersymmetric Yang-Mills theory lives on the 4-dimensional boundary to type IIB string theory with 5 extended and 5 compact dimensions. This boundary theory, the N=4 supersymmetric Yang-Mills theory, has some features in common with Quantum chromodynamics, which describes how hadrons are quarks confined by gluons. For example, a baryon, is a hadron composed of 3 quarks. A proton is 2 up quarks and 1 down quark. A neutron is 2 down quarks and an 1 up quark. A meson is not a baryon, however, a hadron composed of 1 quark and 1 antiquark.