Hopefully this can help clear up some common questions. In a place or two I have expressed uncertainty of survival of descendants, due to lack of resources.I have only listed them from William I to Richard III (last Plantagenet king) but may include more at a later date. If you have anything to add, please feel free!
WILLIAM THE CONQUEROR: None acknowledged, and no others verified. William Peverel, often called his son, was actually probably not.
WILLIAM II RUFUS: None known or acknowledged. Probably a homosexual.
HENRY I: Many, many children by many, many mothers. Some notables include Sybil (she married Alexander I, King of the Scots & was his daughter by Sybil Corbet), Rainald, Earl of Cornwall (also by Sybil), Elizabeth (unknown mother; she married Fergus of Galloway), Eustacie (unknown mother, she married William III, Lord of Montmirial), Maud, Abbess of Montivilliers (prob.by Isabella de Beaumont), Matilda (unknown mother, she married Conan III, Count of Brittany) and Robert de Caen, Earl of Gloucester who's mother is unknown but who's descendants played a big role in English politics for centuries to come.
EMPRESS MAUD: None known or acknowledged. She also never married the Count of Brittany (her illegitimate half-sister Matilda did though, which accounts for the mixup) and it is really very unlikely any of her sons were by Stephen, despite what some romance novels would have you believe.
STEPHEN I: Several by Dameta, but no known descendants from them. His legitimate son, Eustache, had a bastard daughter named Eustachie who became the grandmother of the Count of Ponthieu.
HENRY II: Nearly as many as his grandfather and namesake. The most famous is William de Longspee (who's mother was NOT Rosamund Clifford, but a "Countess Ida"). Another son was Archbishop of York.
RICHARD I: Probably a homosexual, Richard did have one recorded bastard son, Philip of Cognac. Philip's most notable achievement was killing the man who shot his father through the chest with an arrow. After that, he sold some land to his uncle John and disapeared from history. No known descendants. It is even questioned by some historians about whether Philip was his or not.
JOHN I: A daughter, Joan of England, married Llywelyn ab Iorwerth, Prince of Wales. Her mother was probably a Frenchwoman named Clemence. Another son, Richard Fitzroy, married the heiress of Dover and has descendants living to this day. Richard's mother was a daughter of Hamelin de Warrenne and therefore John's first cousin (Hamelin and Henry II had the same father, Geoffrey of Anjou).
HENRY III: None known or recorded.
EDWARD I: None known or recorded; John de Botetourte is generally disbelieved to be his.
EDWARD II: One illegitimate son, Adam, who accompanied him on his Scottish campaigns. Adam died at the tender age of ten and seems to have had no siblings.
EDWARD III: Anne Perrer's children are probably not his; if any are, it's John de Southerey, who's descendants, if any, are unknown to me.
EDWARD, THE BLACK PRINCE: Not a king, but pretty important. One recorded bastard son, Roger de Clarendon, who was later executed and who seems to have had no children.
JOHN OF GAUNT: Also not a king, but ruled England for several years as one in everything but name. Four illegitimate children by Katherine Roet Swynford, who were then legitimized by their parent's marriage. One son, Henry Beaufort, while a Bishop, had a love affair with Lady Alice Fitzalan that resulted in a daughter, Jane Beaufort. Jane married Sir Edward Stradling and was the mother of one son, Sir Henry Stradling. While her husband had numerous illegitimate progeny, Henry was her only child, and all her descendants are through him.
RICHARD II: No recorded children at *all*, either illegitimate or otherwise. He was probably sterile.
HENRY IV: No proven offspring; all legitimate descendants died out, except for two illegitimate grandchildren by his son Humphrey, Duke of Gloucester. Don't know if they left progeny.
HENRY V: See above notes on his father, Henry IV. Only known child was Henry VI, who was an infant on his father's death.
HENRY VI: So saintly even his legitimate son was doubted to be his! Bastards are never reputed, and are frankly unlikely. When his son Edward died, so died the Lancastrian house.
EDWARD IV: Three illegitimate children; Arthur, Viscount Lisle and Elizabeth may have descendants, but I am unsure.
RICHARD III: Several acknowledged illegitimate children, but none are known to have descendants to my knowledge.