Here's a take on it:
Following poor sales in the Batman titles (OTL), DC Comics decides to kill off the character of Bruce Wayne during Crisis on Infinite Earths. Immediately following was one of the most celebrated Batman runs since the titles release.
Firstly was The Dark Knight Returns, following Dick Grayson as he accepts that he is no longer a boy, but a man and that he must accept the title of Batman.
This is followed up with Batman: Year One, detailing Dick's first year in the cowel.
In 1988 Batman: The Killing Joke debuts, following Dick as he tries to rescue his girlfriend (and Batgirl) Barbara Gordon from the clutches of the Joker. This is where Joker discovers that the new Batman isn't the one he's been fighting all these years.
Later that year, A Death in the Family shows the near death of Jason Todd at the hands of the Joker. Following this, Dick forces Jason to retire from the role.
Todd would disappear from comics until 1993's Knightfall, as Dick begrudgingly accepted Todd as a temporary replacement. Dick soon returned to the role, while Todd took a turn on the Titans, the Outsiders, and finally the Justice League.
Dick is eventually killed fighting Darkseid in 2008, and what follows is a battle between a teenaged clone of Bruce Wayne, Jason Todd, and others for the cowl. Eventually Jason takes over as Batman, making Bruce his Robin so that he can be trained like his previous incarnation.
This lasted roughly until the New 52 reboot, wherein Bruce Wayne is now Batman again, Jason Todd has become Agent 37, and Dick Grayson (back from the dead!) stars in Nightwing and the Outlaws featuring Starfire and Arsenal as they try to uncover the mystery of Dick's resurrection. Tim Drake has been recently introduced to serve as the latest Robin.