The dark comedy-drama is written and created by Vicky Jones, Waller-Bridge’s long-term collaborator, who directed the original stage version of Fleabag.
In Run, Merritt Wever (Unbelievable) plays Ruby Richardson, a woman who abandons her settled suburban life to travel across the country with college sweetheart Billy Johnson (Domhnall Gleeson).
Waller-Bridge also has a role in the series, playing Laurel, a woman whom the pair encounter on their travels.
Acclaimed BBC sitcom Fleabag aired its second series in 2019, which its creator has confirmed would be its last.
Waller-Bridge has also written and produced the dark comedy / spy thriller Killing Eve for BBC America, although she handed over writing duties to Emerald Fennell for series two.
She is credited as a writer on the upcoming James Bond movie No Time to Die, leading to speculation that the film will have more of a feminist edge than its predecessors.
Run will debut on HBO at 10.30pm on 12 April, and will premiere in the UK on Sky Atlantic on 15 April.