Who knows that Anakin Skywalker became Darth Vader and that Luke and Leia are his twin children? And when?

22 BBY
R2-D2, C-3PO and the Naboo holy man - know Anakin and Padmé are married [AOTC]

<19 BBY
Panaka - knows about Padmé and Anakin's marriage and later tells Palpatine [ROTSVD]

19 BBY
Obi-Wan, Yoda, R2-D2 - know all [ROTS]
C-3PO - knows all but then his memory is wiped [ROTS]
Emperor Palpatine, Darth Vader - know all except that the twins are alive [ROTS]
Bail Organa - knows about Padmé and the twins being born and that Anakin is the father [ROTS]
Polis Massans and medical droids - know about Padmé giving birth to twins [ROTS]
Some of Palpatine's Shock Troopers, droids at the surgical reconstruction facility - know Anakin becomes Vader [ROTS]
Qui-Gon Jinn's spirt - knows all [ROTS]

18 BBY
Bail Organa - Convinced that Vader is Anakin which he tells Raymus Antilles [DL]
Roan Shyrn - Learns Vader's identity

0 ABY
Vader - learns Luke is alive [VQ]

3 ABY
Luke - learns Vader is his father [ESB]
Prince Xizor - Already knows that Vader is Skywalker and figures out that Luke is his son [SOTE]

4 ABY
Leia - confirms that Luke is her brother and learns that Vader is their father
Han - Leia tells Han that Luke is her brother [RJ]

4.2 ABY
Luke tells Shiro and Lumiya that Darth Vader was his father. [MCSW]

<4.5 ABY
Mon Mothma, Jan Ors, Kyle Katarn - know that Luke and Leia are siblings. [DFRA][DFJK]

10 ABY
Noghri - realize that Leia is Vader's daughter [HE]

11 ABY
Corran Horn and other students at the Jedi academy - learn that Luke and Leia are Vader's children [IJ]

16 ABY
New Republic politicians - In the wake of the revelation of Luke and Leia's Darth Vader ancestry, Senator Peramis brings up the issue during a debate over the 5th Fleet. [BS, EC2]

SOURCE CODE KEY
AOTC Attack of the Clones
BS Before the Storm
DFJK Dark Forces III: Jedi Knight graphic novel
DFRA Dark Forces II: Rebel Agent graphic novel
DL Dark Lord: The Rise of Darth Vader
EC2 The New Essential Chronology
ESB The Empire Strikes Back
HE Heir to the Empire novel
IJ I, Jedi novel
MCSW Marvel Comics Star Wars
RJ Return of the Jedi
ROTS Revenge of the Sith
ROTSVD Revenge of the Sith Visual Dictionary
SOTE Shadows of the Empire novel
VQ Vader's Quest comic

And so the list grows... Thank you for everyone who sent in comments, especially those with page numbers. Keep 'em coming!

QUOTES TO PONDER:
VADER: He is here...
TARKIN: Obi-Wan Kenobi! What makes you think so?
VADER: A tremor in the Force. The last time I felt it was in the presence of my old master.

AUNT BERU: Luke's just not a farmer, Owen. He has too much of his father in him.
OWEN: That's what I'm afraid of.

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Who Knows Which Secrets? thread