In the movie, Bolt is very protective of Penny--the two are actors in an action TV show, but Bolt believes that the whole thing is real and that he must protect Penny from all the bad guys who are out to get her. When Bolt is lost from the studio, his number one priority is to find Penny, even willing to travel across the whole country to do so.
Even after finding out the truth about the TV show, Bolt still wants to find Penny, due to the fact that she is still his best friend. When he returns to the studio he sees Penny hugging Bolt's actor replacement and doubts whether she still cares about him, but the truth is that she misses Bolt as well. The two are apparently so close that Bolt is willing to dash into a burning building to save Penny, who is trapped inside.
Throughout the movie, Penny's only wish is to spend time with Bolt outside of their acting career. At the end of the film, she finally gets her wish and the two of them are seen playing together, along with Mittens and Rhino.
Most people see this pairing as a love between a girl and her dog, but some fans see it to be more romantic. In this case, they may sometime use a human version of Bolt, much like Toothcup.
Note: Please only add examples that are part of the RotBTD fandom.