He had many illegitimate children, including Robert, Earl of Gloucester

From "Debrett's Kings and Queens of Britain" by David Williamson, ISBN 0-86350-101-X, p. 47-49:
Acceded 2 Aug 1100. Crowned at Westminster Abbey on 6 Aug 1100. He had a son born prematurely about Jul 1101 who died in infancy. He was the only son of William 'the Conqueror' born in England. Born at Selby in the autumn of 1068, his mother having accompanied his father on his expedition to subjugate the north of England. Like many youngest sons, he became his mother's favorite and on her death in 1083 she left him her English estates, which, however, he was not allowed to hold during his father's lifetime. Meanwhile he is reputed to have acquired a good education, learning to read and write Latin and also studying English and English law. Perhaps this was with a view toward his entering the Church, often the destiny of youngest sons. His learning earned him the sobriquet of 'Beauclerc' (fine scholar) of which he became very proud, and in later life he declared that 'an unlettered King is but a crowned ass.'
Henry was knighted by his father at Westminster on Whitsunday 1086 and after the King's death the following year he became one of those barons who suffered from the Conqueror's decision to leave Normandy to Robert and England to William. Until Robert resigned Normandy to William in 1096, Henry was constantly being forced to choose between his two overlords and brothers, and whichever side he came down on, he was likely to annoy the other.
Once England and Normandy were reunited under William Rufus, Henry was able to serve the King, and he was fortuitously on hand in the New Forest on the day his brother was killed on 2 Aug 1100. The following day, after William's burial at Winchester, such councillors as were at hand elected Henry king and, after securing the treasury, he immediately left for London, where on 6 Aug he was crowned in Westminster Abbey by Maurice, Bishop of London.
Henry's first act as king was to issue a charter promising a return to his father's ways, and to restore Anselm to the Archbishopric of Canterbury. His next was to seek a wife and his choice very expediently fell upon Edith, renamed Matilda in honor of his mother, the elder daughter of Malcom Canmore, King of Scots by St Margaret, the granddaughter of King Edmund Ironside. He thus reinforced the strain of Saxon blood in the royal family.
The vexed question of lay investiture of ecclesiastical estates threatened relations between Church and state for several years. Anselm refused to do homage to the King for the archiepiscopal estates, claiming he held them from the Pope. The King would not give way and Anselm was deprived of his fiefs and again forced into exile. A compromise was reached in 1107 when the King's sister Adela, Countess of Blois, suggested that the bishops should pay homage for fiefs held of the King, who in his turn would allow clerical investiture.
Henry was a wise ruler, a good judge of men, and a skilled diplomat. The affairs of Normandy occupied the early years of his reign. Robert, his brother, had returned from the Crusade but proved such an ineffective ruler that his barons revolted and invited Henry to come to their aid. Robert was taken prisoner at Tinchebrai and Normandy passed under Henry's rule. More troubles in Normandy and war with France continued to occupy the next few years. In 1109 Henry's foreign policy triumphed in the betrothal of his only legitimate daughter Matilda to the Emperor Henry V, the marriage taking place in 1114.
In 1120 Henry's only legitimate son, William, was tragically drowned with his entourage in the wreck of the White Ship when returning from Normandy. Henry, it is said, never smiled again. Queen Matilda had died in 1118 and in 1122 Henry took a 2d wife, Adeliza, daughter of Godfrey, Count of Louvain, but the marriage was childless. At Christmas 1126 he designated his daughter, the widowed Empress Matilda, as his successor and the following May he chose a 2d husband for her, the young and handsome Geoffrey, son of the Count of Anjou, who was 10 years or more her junior.
Henry was continually traveling from England to Normandy and back throughout his reign. He left England for the last time on 1 Aug 1135. At the end of Nov Henry was at his royal hunting-box at St Denis-le-Fermont, near Gisors, where, says Henry of Huntingdon,'he devoured lampreys, which always disagreed with him, though he was excessively fond of them, and when his physicians forbade him to eat them the King did not heed their advice.' A severe case of ptomaine poisoning followed and Henry died on 1 Dec. He was 67, far short of the 80 years attained by his eldest brother Robert who had ended his days in prison in Cardiff Castle a year earlier.
Henry's body was brought back to England and interred in Reading Abbey, which he had founded. No trace of his tomb remains today, the site being covered by a car park.
Henry brought together the Woodstock menagerie, the first English zoo.
Henry's marriage to Matilda produced only 2 surviving children, that to Adeliza none, but a number of mistresses bore him a large illegitimate progeny, several of whom made a mark in the world, especially Robert, Earl of Gloucester, who was to play such an important part in the next reign.

Reigned 1100-1135. Duke of Normandy 1106-1135. His reign is notable for important legal and administrative reforms, and for the final resolution of the investiture controversy. Abroad, he waged several campaigns in order to consolidate and expand his continental possessions. Was so hated by his brothers that they vowed to disinherit him. In 1106 he captured Robert and held him till he died. He proved to be a hard but just ruler.

Henry I was the fourth and most capable son of William the Conqueror and Matilda, born 1068, and nicknamed "Beauclerc" (fine scholar) for his above average education. He married Eadgyth (who later took the name Matilda), daughter of Malcolm III of Scotland, who bore him two sons and a daughter. One son died very early, and the other, William, died in the wreck of the White Ship in November 1120, leaving the daughter, Matilda, as the sole heir. Eadgyth died in 1118, and Henry married Adelaide of Louvain, but the union produced no offspring. Henry also had two fairly significant illegitimate children - Robert de Mellent, Earl of Gloucester, and Sibylla, wife of the Scottish King Alexander I. Henry's was the longest reign of the Norman line, lasting thirty-five years.
The first years of Henry's reign were concerned with subduing Normandy. His father divided his kingdoms between Henry's older brothers, leaving England to William and Normandy to Robert. Henry inherited no land, but received £5000 in silver. He played both sides in his brothers' quarrel, leading both to distrust Henry, and sign a mutual accession treaty barring their brother from the crown. Henry's hope arose when Robert went on the First Crusade; should William die, Henry would be the obvious choice. Henry was in the woods hunting on the morning of William's death, August 2, 1100. He moved quickly and was crowned king on August 5, his coronation charter denouncing William's oppressive policies and promising good government. Robert returned to Normandy a few weeks later, but escaped final defeat until 1106, at the Battle of Tinchebrai. Robert was captured and lived the remaining twenty-eight years of his life as Henry's prisoner.
Henry was drawn into controversy with the Church over the lay investiture issue - the practice of selling clergy appointments by the king to gain revenue, heavily opposed by Gregorian reformers in the Church. He ignored the situation until he was threatened with excommunication by Pope Paschal II in 1105, reaching a compromise with the papacy: he would officially denounce lay investiture, but prelates were to continue to do homage for their fiefs. In practice, it changed little - the king still had the deciding voice in appointing ecclesiastical offices - but it a marked a point when kingship was viewed as purely secular, and subservient to the Church. A solution to the lay investiture controversy and conquest of Normandy were accomplished in 1106, allowing Henry to expand his power.
Henry mixed generosity with violence in motivating allegiance to the crown, appointing loyal and gifted men to administrative positions. Roger of Salisbury, the most famous of Henry's servants, was instrumental in organizing a department for collection of royal revenues, the Exchequer. The Exchequer quickly gained notoriety for sending out court officials to judge local financial disputes, weakening the feudal courts controlled by local lords, and won the title "Lion of Justice".
The final years of his reign were concentrated on war with France, and succession concerns upon the death of his son William in 1120. The marriage to Adelaide was fruitless, leaving Matilda his only surviving legitimate heir. She was recalled to Henry's court in 1125 after the death of her husband, Emperor Henry V of Germany; Henry forced the barons to swear they would accept Matilda as Queen upon Henry's death. She was then forced to marry the sixteen-year-old Geoffrey of Anjou (founder of the Plantagenet dynasty) in 1128 to continue the Angevin alliance. The marriage was unpopular with the Norman barons, but Matilda and Geoffrey produced a male heir, prompting Henry to force another oath from the barons in support of Matilda.
In summer 1135, Henry refused to give custody of certain key Norman castles to Geoffrey, as a show of good will, and the pair entered into war. Henry's life ended in this sorry state of affairs - war with his son-in-law - in December 1135. Source: www.britannia.com