Randall is Penny's brother. Apart from Penny, he has another unnamed sister, and he is the only son of Wyatt and Susan. He has finally appeared in season 10, but was previously only mentioned.
In "The Dumpling Paradox", Penny says that her friend Christy Vanderbel was engaged to her (Penny's) cousin while she was sleeping with her brother.
In "The Pirate Solution" (S03E04), it is revealed that he works in a meth lab and is described as a 'kind of chemist' and Penny told Leonard that Leonard would like her brother.
In "The Rhinitis Revelation" (S05E06), Penny asks God to make her brother stop cooking methamphetamine but without involving the police when Sheldon's mother decides to get them to pray at a Roman Catholic church.
In "The Status Quo Combustion" (S07E24), Penny wants to schedule her wedding 12 to 18 months away so her brother can attend after he gets out of prison.
In "The Bachelor Party Corrosion" (S09E03), Penny's father tells Penny that her mother is out with her brother celebrating one month of being sober. He had also received his 10th one-month-sober chip.
It's revealed in "The Convergence-Convergence" (S09E24) that Penny's brother had been released from prison due to overcrowding, making him eligible to attend her redo wedding ceremony with the family.
In "The Conjugal Conjecture" (S10E01), Randall comes to Pasadena with his parents to attend the re-wedding, he smiles a lot and embarrasses his mother.
In "The Emotion Detection Automation", Penny mentions that she is trying to get him a job interview and needs a place to stay, which makes Leonard upset and causes a fight between him and his wife.