The Linear Men are "Time Police" who try to keep the past and future safe from careless time travelers. Leader Matthew Ryder discovered the Vanishing Point realm outside of space and time where the Linear Men now maintain their base. Due to the physical stresses of time travel, pieces of the Linear Men's bodies have been replaced with Cyborg limbs. The Linear Men played a key role in the fight against Gog, revealing the existence of a "hyper-time" containing all possible alternate time streams.
After Superman defeated Imperiex by sending him back to the Big Bang, the Linear Men attempted to have the Man of Steel tried for crimes against the timeline. They now appear to have shunned their physical forms and appear only as manifestations of the intellect.
During the story arc of 52, Skeets, who was controlled by Mister Mind, tortured and killed Waverider. Rip Hunter also stated he has locked up the Linear Men and that Waverider LIVES was scrawled across his blackboard.
How the Final Crisis, 52, and Countdown story arcs have altered the Linear Men and Waverider further remains to be seen.
Matthew Ryder and Liri Lee are the sole survivors of the Linear Men. It is unknown, if Matthew has the ability to tap into the Waverider persona. The Ultra Humanite, Black Beetle and Per Degaton have captured the two and following Booster and his group through the time stream.