A major focus of the whole 'Death of the Family' event has been the whole 'Does he or doesn't he' tease in both the main 'Batman' title and in all the tie-ins. Does the Joker know who Batman is, and who all his allies really are?
Batman himself seems to think not, even though there is evidence to suggest otherwise. And the evidence that the Joker knows is piling up with every issue and every encounter between the clown and a Bat-family member.
Here we can discuss all the evidence and all the theories for and against...
Let's list out the facts and assumptions vis a vi the identities of each Bat-family member in turn. There will of course be SPOILERS for every DOTF issue/tie-in released so far-
The abduction of Alfred seemed to cinch it, but then again, Bruce proposed a pretty plausible explanation for why the Joker would target Alfred even WITHOUT knowing Batman's identity. Alfred is publicly known as Bruce Wayne's butler and is a major figure in Gotham in his own right...and Bruce Wayne is also publicly known to be Batman's ally and benefactor. And honestly, there's no real reason why that explanation doesn't work as far as the matter of Alfred's kidnapping goes.
Then there's the matter of whether the Joker entered the Batcave or not years ago. Batman is confident that there's no way he could have made it in. Let us suppose however that the Joker DID make it in...does having access to the Batcave itself necessarily reveal to one that Batman is Bruce Wayne? Unless the Joker found a way to access Wayne Manor, or secretly observed Batman unmasking (both possibilities being highly unlikely), there's no way he would necessarily know the truth. Then again, with the Joker, ANYTHING is possible...
Last, but not the least, there's the fact that the Red Hood Gang targeted Bruce Wayne years ago (back when Bruce was fighting them as an anonymous vigilante). We still don't know the whole story behind that, but assuming the Red Hood Gang leader was (pre)Joker, and that he'd somehow tied in Wayne with the anonymous vigilante and later, by extension, with the Batman (who would have debuted not long after)...then again, the Joker could have presumably known all along.
Basically, it could go either way here.
There seems to be a LOT of evidence to suggest that the Joker knows that Batgirl is Barbara Gordon, the woman he once crippled. There's the fact that, of all people, he had her mother abducted. Now, there's a plausible reason why Joker would abduct Barbara Sr. (given she's the ex-wife of Jim Gordon, another target the Joker loves tormenting)...but why would he specifically use her in a plan against BATGIRL of all people?! Unless he knows...
It's not really difficult for the Joker to have put it together. After all, he's been keeping an eye on the Gordon family and knows all about them. He probably figured out that Batgirl disappeared shortly after he crippled Barbara Gordon, and that Batgirl suddenly reappeared years later not long after Barbara's miraculous surgery which put her back on her feet. It would have been easy for him to put two and two together, even if he DOESN'T know Batman's identity.
If Joker doesn't know who Batgirl is...then there needs to be some kind of explanation for why he abducted Barbara Sr...because as far as he knows, there can be no other connection between Batgirl and the woman. Unless its just random...Joker combining his attacks on two targets, by abducting Jim Gordon's wife and using her to play mind-games with Batgirl!
If Joker knows that Batman is Bruce Wayne, figuring out the identity of the current Robin would be a no-brainer, especially since its publicly known (as per Batman Inc) that Damian is officially recognized as Bruce Wayne's son. That apart, there's nothing which really suggests that the Joker knows who Robin is...except for the fact that he constantly refers to Batman as Robin's 'father figure'. Now that of course can be a perfectly innocent and understandable remark; Batman has been a 'father figure' to all the Robins after all; but its curious that he's chosen to employ the phrase with Damian, but hasn't employed it with Nightwing or Red Hood thus far (although he's aware they are both former Robins)...
Its also possible that Joker DOESN'T know Batman and Robin's true identities, but has still somehow deduced that the current Robin is Batman's biological son (maybe based on his age, some kind of similarity in their fighting styles/behavioral patterns etc.)?
Joker certainly knows that Nightwing was the original Robin (though that seems to be common knowledge among the rogue's gallery-even Penguin is aware of the fact, as is Lady Shiva). But does he know that Nightwing is Dick Grayson, Bruce Wayne's former ward?
The fact that he abducted/freed Raya Vestri, known to be an enemy/former lover of Dick Grayson, and has chosen to target Amusement Mile, on the surface clearly suggests that Joker knows. However, there is an alternative explanation...its possible that the Joker abducted Raya knowing her to be a criminal whom Nightwing apprehended. And maybe he's targeting Amusement Mile, current home of the Haly's Circus, because Nightwing previously saved the circus, and thus he's guessed that Nightwing has some kind of an interest in protecting the circus.
This is again something which CAN go either way, though on the surface at least, it seems that Joker knows the truth here.
Red Hood/Jason Todd
If there is even a grain of truth in the revelations of the zero issue, then its evident that the Joker knows EVERYTHING. Not only would he have known all along that Jason Todd was Robin and now Red Hood, but moreover, he could have figured out that Bruce Wayne was Batman (if he hadn't already at that point), and by extension, the identities of everyone else in the family.
Alternatively, its possible he knows SOME things about Jason's past, and has just spun those few facts into an elaborate lie/fantasy, trying to convince Jason (and perhaps himself) that he is the architect of Red Hood's destiny. But how would that work exactly? Its one thing for him to have observed an unknown teenager escaping Leslie Thompkins clinic with stolen drugs and confronting Batman (and from there on, deducing that that teenager subsequently became the second Robin). But him knowing about Jason's father getting shot, or about his mother? That's a whole different ball-game.
Of course, its possible that the Joker knows that Jason Todd was Robin (and everything about him manipulating Jason into becoming Robin was true)...but he doesn't know that Bruce Wayne is Batman. Even if he knew that Jason was taken in by Bruce Wayne, he could simply have assumed that Bruce, being Batman's ally, provided a home for the latter's new sidekick.
Red Robin/Tim Drake
Joker, again, CLAIMS that he knows who Tim is, though Tim doesn't believe him and again, he doesn't really offer much conclusive proof to suggest that he really knows. Interestingly enough, Joker does know that Tim was the 'third son' of Batman, and he also knows that Tim 'left' the Bat-family of his own choice. Of course, that doesn't really prove anything...he obviously knows that the first Robin became Nightwing, and he personally killed the second Robin (who later became Red Hood), so it would be obvious to him that Red Robin was the 'third'. Moreover, he's been keeping tabs on the Bat-family so careful observation would reveal that Red Robin no longer works with Batman in Gotham but is instead busy leading the Teen Titans. So really, there's nothing here to suggest that the Joker knows, beyond his own claim that he does.
Then again, if the Joker knows who Batman is, or knows who Red Hood is (the latter being a very likely possibility, as discussed above) THEN its very likely he knows who Red Robin is. It would be interesting to know what the Joker meant when he said he knew Red Robin's 'real name' though...did he mean 'Tim Drake', or did he mean Tim's unknown original last name?