In current continuity, it has not come up, but based on what we know and what we can assume, and what we can safely port over from old continuities, the real answer is: lots of people. Bruce Wayne is fantastically wealthy, a devoted philanthropist, and arguably the largest private source of funding for vigilantes.
First, the title of Batman will certainly be passed on. There have been plenty of times Bruce Wayne has been knocked out of commission, but the mantle of Batman has never remained empty for long.
Bruce can't bequeath the title legally, so there may be some quibbling over the title, but it's unlikely it would end up with anyone but Damian Wayne, the current Robin and Bruce's son. Primarily because the other Robins have all stepped out of the shadow of the Batman and would support Damian's claim. Also, see below, regarding Wayne Enterprises.
Wayne Manor would possibly go to some sort of trust, but most likely will fall to Damian Wayne, who currently lives there. Even if Bruce's will says otherwise, Damian would have a strong legal claim in most jurisdictions, as it is his primary residence.
As for Wayne Enterprises, it will almost certainly go to one of the Robins, either Damian Wayne or Tim "Red Robin" Wayne, Bruce's adopted son. Dick "Nightwing" Grayson has spent lots of time away from Gotham, so he's not the obvious choice. Jason "Red Hood" Todd has a criminal past. It's unlikely it would be split so company control can be maintained. Also, whoever controls Wayne Enterprises has a stronger long-term claim on the title of Batman, because as Batman, Inc. has demonstrated, the sponsorship of vigilantes worldwide is really the greatest force Batman brings, and that comes from the fantastic wealth and tech of Wayne Enterprises. Whoever controls the gadgets, controls the title.
The rest of Bruce Wayne's fantastic wealth would at least partially be left to setup permanent trusts for vigilantes like Barbara “Batgirl" Gordon, et al. In addition, funding would certainly be left for Gotham Academy and othe charities and pseudo-research institutions that support the education and tooling for the Bat family.