Audio sountracks exist for all the missing episodes, which has allowed the BBC to release the audio soundtracks on CD, with supporting narration. The odd surviving episodes from these stories, known as 'orphaned' episodes, were released on video in the Hartnell Years, Troughton Years, Daleks - The Early Years, and Cybermen - The Early Years compilations, and on DVD as the 'Lost in Time' compilation, both of which included many of the surviving clips from lost episodes.
In addition there also exist still photographs taken off-screen by photographer John Cura. Cura was hired by the BBC, and independently by many actors and production staff, to document the transmission of many of their most popular programmes from the 1940s to the 1960s, including Doctor Who. These 'tele-snaps' were generally used to promote BBC programmes and for actors, directors, and other production crew members to keep a visual record of their own work in the days before home video recorders. In many cases, they form the only visual record remaining of several Doctor Who serials and other missing episodes of many programmes.
You can view the telesnaps online in a 'photonovel' format for most missing episode stories on the Beebs classic series website here. Notable exceptions include Marco Polo, the Daleks Masterplan and The Invasion.
Since the late 1990s, reconstructions of the missing serials have been made by fan groups such as Loose Cannon Productions (visit their website here), who distribute them free. These "recons" are based on the directors' original camera scripts, and use a combination of the surviving soundtracks, surviving footage, photographs, still images (especially Cura's tele-snaps) and specially-recreated material. Although technically infringing copyright, these recons have been tolerated by the BBC, provided they are not sold for profit and are only distributed in degradable, non-digital formats such as VHS.
Officially released high-quality reconstructions using the same methods were made for the BBC Video releases of The Ice Warriors (a 12-minute 'highlights' reconstruction bridging the missing Episodes 2 and 3) and The Tenth Planet (a full reconstruction of the missing Episode 4).
Abridged VHS releases, with the surviving episodes and linking material recorded by actors (The Reign of Terror, The Crusade, and The Invasion) or with audio CD recordings of the missing episodes (The Crusade and The Ice Warriors) were also released, and subsequent DVD releases should see improved formats.
In 2005, BBC Audio released a reconstruction in an "MP3 CD" format. Under the 'Doctor Who: Reconstructed' banner, the CD included the same audio portions as the previous audio CD release, but on a single disc with Macromedia Flash-animated and synchronised slideshow of tele-snaps and other (publicity) photographs. (The surviving clips could not be included.) The Power of the Daleks was the first and last such reconstruction to be released: a mooted release in this form of the following story, The Highlanders, did not go ahead, due to poor sales of the initial release.
In 2006, The Invasion, an eight-episode Second Doctor serial of which six episodes survive in the archives, was released on DVD with the missing Episodes 1 and 4 animated by Cosgrove Hall, matched up with a newly-remastered soundtrack.
The DVD box set Doctor Who: The Beginning consisted of the first three serials and a 30-minute reconstruction of Marco Polo, of which no footage exists.