The Underground is an alternate version of Shibuya, Japan. This is the location for the entire plot of the story. To be allowed into the UG, a person must die in the realground, or real Shibuya, and are offered the chance to enter the Reaper's Game. To do so though, they must sacrifice their most valued possession. If they survive for a week, they are revived in the RG and may commence with their lives.
Characters in the UG are invisible to those in the RG but are capable of influencing thoughts through memes, or simple words or phrases that trigger certain actions.
The Composer is in charge maintaining Shibuya and of all what happens in the UG. The Conductor instructs the Reapers on what the Composer wants done and ultimately has control of the UG for a limited time, a single week.