[This is part of Catholic Analysis’ special series on Pope Alexander VI. This part, the second, deals with the pontiff’s family. Read the first part.]
Vannozza dei Cattanei was Rodrigo Borgia’s mistress and the mother of his children. She “doubtless was of great beauty and ardent passions; for if not, how could she have inflamed a Rodrigo Borgia? Her intellect too, although uncultivated, must have been vigorous; for if not, how could she have maintained her relations with the cardinal?”  He was “faithful through his life to [this] one woman”. After the births of their children and before he ascended to the Papacy, she was put away in luxurious accommodations and received provision for the rest of her life. She “agreed with the Cardinal about the need for her to live a life of retirement rather than presume on a position which would then have given far less scandal than at any other period of history”. She paints a picture of Rodrigo as anything but a “debauched man”. 
Cesare, Rodrigo’s first-born, was a fiery man. Under his father’s patronage, he rose to the rank of cardinal, a position which he later resigned in order to become more involved with military efforts in the Papal States. He is the first person in history to resign the cardinalate.
Giovanni, the militant second-born, served as gonfaloniere (military “standard-bearer”) in the Papal States until his death – likely as a result of a sexual encounter  – in 1497. (NOTE: There is disagreement over the ages of Cesare and Giovanni.)
Lucrezia, the third in line, was exceptionally beautiful, graceful, and precocious. “She had received instruction in the languages, in music, and in drawing, and later the people of Ferrara were amazed at the skill and taste which she displayed in embroidering in silk and gold. ‘She spoke Spanish, Greek, Italian, and French, and a little Latin, very correctly, and she wrote and composed poems in all these tongues,’ said the biographer Bayard in 1512.”  She was, no doubt, good fruit from the Church’s many investments in education up to that point. Out of all of Rodrigo’s children, she lived the longest: until 1519, at the age of 39.
As for Gioffre, little is known of him. Overshadowed by his brothers, his place in history has been a tad undermined. He was mild-mannered.
There are more than a few that raise objections to the fact that Rodrigo broke his clerical vows of chastity, but these persons almost always neglect two key points. First, that he legitimized them and provided for his family, when he could have just discarded them, reveals an immense amount of love and charity. Second, St. Peter, the first pope, is traditionally believed to have been the father of at least St. Petronilla, a first-century martyr.
Some, like writers for The Dublin Review in 1858, dispute Rodrigo’s paternity of his children, saying things like, “..[I]t is admitted that, in certain legal proceedings, sworn depositions attested that [Cesare] and [Lucretzia] were the children of another — we believe of his brother. It is easy to say that the depositions were false, but no one said so at the time; and it is too much to dispose of sworn evidence by ex post facto assertions. The truth is, that it was the custom of [p]opes in those troublous times, to have a relative, generally a nephew, a man of [vigor] and martial prowess, to conduct the defen[s]e of the Papal territories against the rapacious states by which it was surrounded, and who were for ever seeking to spoil it. And the relative, according to papal usage, would be called ‘my son:’ which is probably foundation enough for malignant enmity to base a foul calumny upon. Assuredly just as much foundation exists for the imputation in the case of many a pontiff acknowledged to have been good and holy.” 
Others, meanwhile, try to say that he had more children. Some say three more: Geronima, Isabella, and Pedro Luis, because these three are thought to have Vannozza as their mother.  Others say there may have been even more than that, including Ottaviano. 
The former theory sounds more likely than the latter, at least, but to deny his fatherhood – which has been corroborated by so many – comes off as ludicrous. In fact, Stefano Infessura (an Italian humanist historian) records that Rodrigo, “wishing to make his natural son [Cesare] a cardinal, caused it to appear, by false testimony, that he was the legitimate son of a certain Domenico of Arignano”. So, such testimonies were, in actuality, only to further legitimize his children! 
The second group, alternatively, presents a serious quandary: if these were indeed his, why would Rodrigo not recognize them, seeing as he recognized the others and they would all have been in similar circumstances? Such a possibility seems even more ludicrous.
Anyway, perhaps, on this topic, Christians should exercise charity like Rodrigo did.
1. Ferdinand Gregorovius (translated by John Leslie Garner), Lucretia Borgia
2. Michael de la Bedoyere, The Meddlesome Friar and the Wayward Pope, p. 64
3. Michael de la Bedoyere, The Meddlesome Friar and the Wayward Pope, p. 20
4. Ferdinand Gregorovius (translated by John Leslie Garner), Lucretia Borgia
5. The Dublin Review, Vol. XLV, September-December 1858, p. 341-342
6. Michael de la Bedoyere, The Meddlesome Friar and the Wayward Pope, p. 64
7. Christopher Hibbert, The Borgias and their Enemies, p. 30
8. Ferdinand Gregorovius (translated by John Leslie Garner), Lucretia Borgia