Short answers, pulled from the story.
Shmi Skywalker states that Anakin has no father and was conceived through midi-chlorians rather than traditional means. This discovery marks the beginning of a bloodline with strong inherent capabilities related to the Force.
Padmé Amidala gives birth to twins Luke and Leia shortly after the first Empire Day before dying herself. Obi-Wan Kenobi and Yoda decide to separate the children to keep them hidden from Darth Vader.
By the end of Star Wars: Episode IX, The Rise of Skywalker released in 2019. Rey adopts 'Skywalker' as her surname to honor the family while cutting ties with her Palpatine lineage.
Adam Driver portrays Ben Solo/Kylo Ren in the sequel trilogy starting with The Force Awakens in 2015. His descent causes Han and Leia to break up though neither stops loving their son.
Jaina Solo is the eldest child of Han Solo and Leia Organa, created by Timothy Zahn for the Thrawn trilogy set about five years after Return of Jedi. She becomes a major character in Young Jedi Knights and later duels and kills her twin brother Jacen in Betrayal in 2006.