What are the advantages of four engined aircraft over twin engine aircraft?

Historically, the four engine design was used to ensure performance and reliability. Aircraft engines back in the early years of aviation were prone to failures and mechanical trouble, and with a four engine airplane, losing 1 engine meant the aircraft retained 3/4 of its performance, versus 1/2 with a two engine design. Additionally, engine technology at the time meant that some aircraft actually required 4 engines in order to have sufficient thrust for takeoff/climb (i.e, B-747).

Today, the only real advantage a four engine aircraft enjoys is performance (as mentioned above), and an increased level of redundancy. Additionally, both of the advantages are rapidly dissolving, as no current commercial transports (other than A380 and 747) employ a four engine design.