The new Barbican season also features new work from Ivo van Hove and Simon McBurney
Recent WhatsOnStage Award winner David Tennant will return to the role of Richard II, while Antony Sher and Jasper Britton reprise Falstaff and Henry IV respectively in the Henry IV plays, and Alex Hassell plays Prince Hal/Henry V.
Gregory Doran, RSC artistic director, said: "We have loved being back at the Barbican and I am thrilled to give London audiences the very first opportunity to see all four History plays together in repertoire."
The RSC season, which runs from 7 November to 24 January under the banner King and Country: Shakespeare's Great Cycle of Kings, forms part of the Barbican's programme marking 400 years since Shakespeare's death in 2016.
Other highlights include Toneelgroep Amsterdam's Kings of War (22 April and 1 May 2016), directed by Ivo van Hove, which reimagines Henry V, Henry VI and Richard III as modern-day political leaders.
And Malthouse Theatre's The Shadow King will transport King Lear to the "story-rich and resource-laden terrain" of northern Australia.
'This is a hugely exciting time for the Barbican' (Nicholas Kenyon)
Elsewhere, Complicite will continue their longstanding relationship with the Barbican when they present The Encounter, based on the story of explorer Loren McIntyre's search for the source of the Amazon River in 1969.
Inspired by Petru Popescu's novel Amazon Beaming, the play, which runs from 12 February to 6 March 2016, is directed and performed by Complicite artistic director Simon McBurney.
Told by an Idiot also return to the Barbican with a Christmas revival of their 2013 children's show Get Happy (17-30 December 2015).
The music programme includes The Dark Mirror – Zender’s Winterreise (12-14 May 2016), a "theatrical version" of Schubert’s song cycle featuring Ian Bostridge and the Britten Sinfonia, staged by Netia Jones.
And the film programme includes Beckett on Screen, complementing the Theatre's International Beckett season in June, screening four programmes of 11 rarely shown Samuel Beckett plays.
The Barbican also announced today that its audience numbers for the past year exceeded 1.2million, higher than the 1.1million who attended in the 2012 Olympic year.
Nicholas Kenyon, the venue's managing director, said: "This is a hugely exciting time for the Barbican as we look to the future. The next five years will transform our offer to audiences."