"Try my life! People look at you because your dad’s the famous Harry Potter, saviour of the wizarding world. People look at me because they think my dad is Voldemort."
-- Scorpius Malfoy (CC2.16)
An only child, Scorpius, along with his mother and father, lives apart from the wizarding world due to Astoria’s illness (CC4.4). To the disapproval of his grandparents, Lucius and Narcissa Malfoy, Astoria raised Scorpius to be tolerant and to have respect for Muggles (Pm). However, due to their secluded lifestyle and general assumptions about Malfoy family pride, a persistent rumour arose in the Wizarding World that Scorpius was really the son of Voldemort rather than of Draco (CC1.3, CC1.4, CC4.4).
Although very close to his mother, Scorpius looked forward to going to Hogwarts and making friends his own age (CC2.6). Funny and smart, he and Albus Potter become best friends after their initial meeting on the Hogwart’s Express (CC1.3, CC2.6) and both being sorted into Slytherin House. The loss of his mother during their third year brought Scorpius and Albus even closer together. They are generally considered to be inept misfits (“losers”) by the other students at Hogwarts and they spend all their free time together (CC1.4, CC2.8, CC2.16).
At the start of their fourth year at Hogwarts, Albus talked Scorpius into joining him on a quest to “save” Cedric Diggory. Although reluctant, Scorpius follows Albus off the train (CC1.10) and, after convincing Amos Diggory they are serious, the boys are allied with Amos’s niece Delphi (CC1.14). To prevent Cedric from being killed as the “spare” joint winner of the Triwizard Tournament with Harry Potter (CC1.6, GF32), the solution decided upon is to change history using an Experimental Time-Turner that had been recently confiscated by the Ministry of Magic.
Together with Delphi they steal the Time-Turner from the Minister for Magic’s bookcase (CC1.19) and then go back in time to prevent Cedric from scoring during the First Task (CC2.6, CC2.7, GF20). Their success does change past history, but not in the ways they expect (CC2.9). They make a second attempt, trying to humiliate Cedric during the Second Task (CC2.20, GF26), but this further use of the Time-Turner causes history and therefore their present day lives to turn even more horribly wrong (CC3).
With Scorpius and Albus left stranded in time by Delphi during the Third Task (CC3.20, GF31), an alliance formed of the grown-up parents (Draco Malfoy, Harry Potter, Ginny Weasley, Hermione Granger and Ron Weasley) uses the Golden Time-Turner and works together with Scorpius and Albus to ensure that time and history remain unchanged. The relationship and understanding between Scorpius and his father then improves, as does that of Albus and Harry (CC4).
Other canon notes and references
Scorpius attracts nicknames. Some names he calls himself: "Scorpius the Dreadless" and "Malfoy the Unanxious" (CC3.14) and others are given to him: "The Scorpion King" (CC3.2, CC3.4, CC3.5, CC4.14) and "Bread Head" (CC3.14). The mean and hurtful epithet "Son of Voldemort" is written by other students on his trunk (CC1.10).
Scorpius (the Latin form of Greek: Σκορπιός/Skorpios) means "scorpion" (BabyNamespedia). It is the name of one of the constellations and Scorpio is also the astrological sign for birthdays falling between 23 October and 21 November (Wikipedia). The scorpion is a predatory arachnid which has a nasty sting in its tail (Wikipedia). Other members of the Malfoy and Black families are also named after constellations or stars.
From the Web
Pottermore: Harry Potter and the Cursed Child
The theatre production opened in the West End of London at the Palace Theatre on 31 July 2016, with Anthony Boyle in the role of Scorpius Malfoy: Harry Potter and the Cursed Child, Parts One and Two