Mass. Adding more mass to a wall or ceiling in the form of additional drywall or concrete blocks is a very good way of reducing sound transfer, especially low wave length sound. Unfortunately this method is very expensive, eats into square footage, and often not feasible.
Separation. Building a secondary wall or ceiling is also an extremely effective way to reduce sound transfer. Secondary walls made of steel studs (great for sound) and 5/8" drywall (mass) will reduce sound transfer dramatically. Like adding mass, this method is very expensive, requires additional space, and often not feasible.
Absorption. Adding a sound absorbing material like foam insulation to a wall or ceiling cavity greatly reduces the "amplification" effect of sound within a cavity. This method is the most non-invasive method of reducing sound transfer. It does not require additional square footage, and is nearly always a feasible option.
The scope of a sound reducing project, whether simply adding foam to a wall, or adding foam & building a secondary high mass wall, depends on the existing sound levels and the desired sound levels after the project is completed.