As students of the story of Richard III know, part of the reason he claimed the throne over the heads of Edward IV's two young sons (also known as the Princes in the Tower), was because he believed that Edward was not the son of their father, Richard Plantagenet, and hence not a genuine king. Which meant that Edward's sons were not eligible to be kings either.
So how true is the story?
The mystery goes back to the birth of Edward in Rouen, France in 1442. Richard Plantagenet was away on campaign for five weeks, in the period that Edward would have been conceived, and stories abounded that Edward's real father was an archer who worked in the palace housing Cecily Plantagenet, Edward's mother. The rumours were strengthened because Edward looked nothing like his father, Richard Plantagenet, and nothing like his brother Richard (who became Richard III).
Judge for yourself. Here is Richard Plantagenet:
Richard Plantagenet died in the battle of Wakefield in the War of the Roses, along with his second son Edmund, leaving behind three sons, Edward, George and Richard. The three brothers continued the fight, and in 1461, Edward won a crucial battle against the Lancastrians, with the help of a cousin on his mother's side, the Earl of Warwick, and was crowned King.
This is where it gets interesting.
With the help of Warwick, Edward starts to kill the men who were on campaign with Richard Plantagenet back in 1442, when he was supposed to have been conceived. Those men would have been the only ones able to testify that Richard Plantagenet was away from his wife Cecily at the crucial time. John de Vere, earl of Oxford, and James Butler, earl of Ormonde and Wiltshire were executed in the first three years of Edward's reign. Another of the men who had accompanied Richard Plantagenet in 1442 was Sir Richard Wydeville, later Earl Rivers, Edward IV's father-in-law, whom he had executed in 1469. What Edward's wife Elizabeth thought about her father getting executed is not recorded.
It got even more deadly.
In 1478 Edward's brother George, Duke of Clarence, made a claim for the throne, on the grounds that Edward was not Richard Plantagenet's natural son. Edward could not kill his brother directly without the consent of Parliament, so he put his brother on trial before Parliament, and got Parliament to pass a bill of attainder stripping George of his rights of succession. George was then killed, to the distress of their mother Cecily.
Cecily's pain may have been deeper because George only moved to challenge his brother after Cecily herself, in a rage when she found out that Edward had secretly married Elizabeth Wydville in 1464 without telling her (or inviting her to the nuptials), is said to have declared he was unworthy of being king and she was minded to tell everyone he was illegitimate and have him removed from the throne.
So Edward IV killed his brother George and his father-in-law Sir Richard Wydeville to protect his right to the throne
But what about Richard?
With George killed, Richard becomes the only natural son of Richard Plantagenet. Here is the image produced from a reconstruction based on his skull:
He does look like his father Richard Plantagenet, but not like his half-brother Edward.
Richard appears to have had genuine affection for Edward, they grew up together after all, and he refused to move against him.
But when Edward IV died in 1483, Richard claimed the crown to become Richard III, and Parliament hearing the evidence, decided to declare Edward IV's children illegimate. Parliament was also swayed by testimony from a priest that Edward IV was a bigamist, who had been married to Eleanor Butler, at the time he secretly married Elizabeth Wydeville in 1464 (which meant all Elizabeth's children were illegitimate). The secret marriage to Elizabeth, remember, is the reason Cecily Plantagenet made her outburst that Edward was illegitimate and set the events in train.
Richard III only lasted two years on the throne. He was killed in battle on Bosworth field by a complete upstart, Henry Tudor, who fought with French mercenaries and who had no claim to the throne. To legitimise himself, Henry Tudor persuaded Parliament to make Edward IV's children legitimate again, and then married Edward's eldest daughter Elizabeth, which whom he founded the Tudor dynasty.
Of course if Edward IV's two sons were alive, then making Edward's children legitimate again would mean they claimed the throne ahead of Henry Tudor, and it is thought Henry had them killed in the Tower.
Medieval times were very bloody.