From episode 8 of season 4 of The Blacklist -- Dr. Adrian Shaw: Conclusion -- we've supposedly learned that Raymond Reddington is, by his own admission, Elizabeth Keen's father.
As much as we'd all like to believe this, I think there are some crucial points being overlooked that should tell us otherwise.
First and foremost though, the issue of DNA tests should be addressed. Many people are stating that a simple DNA test would (dis)prove the relationship between Liz and Red. Unfortunately, this is false as DNA tests are irrelevant. Please hear me out before you say I'm crazy.
If you recall back to season 1 episode 12 -- The Alchemist -- Liz says to Red, "You can't change DNA."
Red's response was, "Well, the Alchemist can."
In my opinion, these two lines were the most crucial things ever said during the show in its entirety. I probably don't have to explain why, but I will just so there's no confusion.
It has now been "confirmed" by DNA test results that Alexander Kirk is not Lizzy's father. However, as we learned during S1E12, DNA can be altered. That's not to say that DNA is necessarily being altered now, but if the Alchemist was able to alter anyone's DNA, it's anyone's guess as to whether or not he altered that of Liz, Red, Kirk, etc., nor what his motives were, nor when he did so (if at all).
That aside, it has now even been "confirmed" by Red himself (albeit no more than a verbal confirmation while at gunpoint) that he (Red) is Lizzy's father. Two points to be made here, then I'm done.
First is probably the more obvious point: at the end of S1E10 -- Anslo Garrick: Conclusion -- Reddington tells Liz that he is not her father. See this article for more insight on that topic.
The second is probably equally obvious, but it's this simple: Red repeatedly assures Liz that he has never lied to her. If that is true, then what he told her at the end of S1E10 is by default also true, and therefore Raymond Reddington is not Elizabeth Keen's father.
Feel free to correct me if I'm wrong on anything you've just read, but either way I think I speak for most fans of The Blacklist when I say that this whole never-ending question of whether or not Red is Lizzy's father is really starting to lose its appeal.