The Coleman-Mandula Theorem

Intuitive

Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.

Concrete

The Coleman-Mandula Theorem by Sascha Leonhardt

Abstract

Loopholes

For a nice discussion, see Section 5.2 "Living without Supersymmetry" in New Symmetry Concepts for Spacetime and Unification by Sebastian Ohmer.

There are also symmetries that are not captured by the theorem at all. These symmetries are for example discrete symmetries and spontaneously broken symmetries. The latter being symmetries that do not act on the S-matrix level and therefore evading the requirement to commute with S. Rather, these symmetries are symmetries of the action of the theory and are spontaneously broken before any calculation.The Coleman-Mandula Theorem by Sascha Leonhardt

Why is it interesting?

The Coleman-Mandula theorem is often invoked as a motivation for Supersymmetry.