What is the best way to soundproof a solid wall?

The best way to soundproof a solid wall follows three main principles:

Increase the mass of the existing wall. The heavier and more dense the wall is, the harder it is for airborne sound to pass through it. Use different high density materials, as different materials are better at blocking different sound frequencies. It is the law of diminishing returns if simply using more of the same material
Add resilience to the wall. Soundproofing needs to be able to flex to dampen and absorb sound vibrations
Decouple the soundproofing from the structure. Sound energy can travel more easily through connected surfaces and decoupling from the original structure (using a system like the ReductoClip System) greatly increases the sound reduction performance