Batcave (Arkhamverse)

The Batcave is an underground cave system located below Wayne Manor and is home to Batman. Although the Batcave beneath Wayne Manor is Batman's main base of operations, there are also other Batcaves in Gotham City, including one beneath Arkham Island.

Pre-Arkham
When he was a young boy, Bruce Wayne found an underground cave system located below his home in Wayne Manor and it would be useful when his parents died and Wayne swore to fight crime. It would be his secret base of operations as Batman. It houses his Batmobile, his Batwing, his Batcomputer and Batman's many other gadgets.

Arkham Asylum Batcave
Batman eventually created a small Batcave under Arkham Island and stored some supplies there just in case a massive breakout would occur in Arkham. That breakout would eventually occur when Joker broke out of Arkham for use of Titan and planned to eventually take control of Gotham. The Arkham Batcave would eventually be useful for Batman to look up the Titan formula and eventually defeat Joker. During that time, Joker attempted to take control of the Batcave by using his thugs but thankfully, they were defeated by Batman. The Batcave would eventually be destroyed by Poison Ivy with her plants destroying the Batcave. The Batcave, the sewers near it, and Arkham Island would eventually be taken over by the GCPD.

After Arkham
After the Arkham events, Batman (as Bruce Wayne) would eventually by incarcerated in Arkham City. Hugo Strange's TYGER guards would attempt to take over Wayne Manor and the Batcave beneath it to take advantage of Wayne being in Arkham. Luckily, Nightwing would eventaully take out the TYGER guards and keep Wayne Manor and the Batcave safe.

Trivia

 * The Batcave would be featured in the Nightwing Challenge Maps DLC in Batman: Arkham City.