The Residents - The Mark of the Mole and the rest of the Moleshow albums, etc.
Van Dyke Parks - Tokyo Rose
Peter Hammill -* Nadir’s Big Chance*
All these posts and no one has mentioned the arguably first concept album? The Beatles St. Peppers Lonely Heart Club band?
…and you call yourselves old Hippies! 
Thick As A Brick - Jethro Tull
Selling England By The Pound - Genesis
Jeff Wayne’s Musical Version of The War of the Worlds.
Layla and Other Assorted Love Songs by Derek and the Dominoes. It’s all about Eric Clapton’s relationship with Patti Harrison.
The Who Sell Out
Rick Wakeman’s The Six Wives of Henry VIII
Somebody beat me to Willie’s LP’s. But I’ll add another by Johnny Cash: Sings the Ballads of the True West. Done while he was in a dangerous period. (He had several.)
Anthems in Eden is actually half of one of those old LP things, but it definitely has a concept. British folkies Shirley & Dolly Collins recorded a set of traditional tunes with backup by the Early Music Consort. The overall theme was the end of traditions brought by the Great War–still in living memory in those days. The novel musical setting supposedly inspired the later British folk-rockers…
Spirit’s 12 Dreams of Dr. Sardonicus
ELP’s Brain Salad Surgery
Lots of great music here.
I’ll add:
10cc - How Dare You
Tubes - Prime Time
Pet Sounds (Beach Boys) was out 13 months before Sgt. Pepper, and greatly influenced it.
Freak Out! was before Sgt Pepper by almost a year.
Face to Face by the Kinks beat it by almost 7 months
Leaving the Rock genre, Woody Guthrie’s Dust Bowl Ballads was put out in 1940.
*A Passion Play
Dark Side of the Moon, Wish You Were Here, Animals
Tommy
I Robot*
Since no one else will
Kendrick Lamar - Good Kid, M.A.A.D City
Seriously one of, if not the best, concept album to come from hip hop and a very tight, focused concept album at that. Amazing album from someone who will probably go down as one of the Greatest of All time (GOAT!)
Edit: plus the album just absolutely bangs. https://www.youtube.com/watch?v=w1pzbhv4EVw
Much as I love the album, the title is the only thing that’s even remotely like a concept, and it’s no more a concept than Randy Newman’s 12 Songs is. Considering the group was falling apart as they recorded, it’s unlikely that there was any real connection (other than a couple of segues).
I’m not trying to hijack the thread, but related:
Has anyone done a 7 Sins concept album? It seems like such an obvious theme, but I can’t think of anyone who’s done it.
Moody Blues Days of Future Passed
I always assumed a connection between the album and the 1961 horror movie Mr. Sardonicus. The lyrics of several of the songs effectively mirror situations or attitudes in that movie, for instance:
Nothing to Hide - “we’re married to the same bride, she eats away from inside” reflects the fact that both Sardonicus and the young doctor are in love with the woman, and they are both concerned with Sardonicus’s disease.
Nature’s Way - Sardonicus is dying.
Why Can’t I Be Free? - the doctor is unable to cure Sardonicus.
and so on.
I don’t think Face to Face quite counts; but Arthur and Village Green are both great concept albums, as are the Kinks’ Lola vs Powerman and the Moneygoround, Schoolboys in Disgrace, and (first in quantity though not in quality) Preservation Acts 1 & 2.
Maybe also The Who Sell Out?
Parks has several concept albums; my favorite is Jump!
I think my overall favorite concept album is Daniel Amos’s Songs of the Heart.
Yep. Not exactly mainstream, to put it mildly, but a pretty damn good album.
Best? Hmmm…I’ll throw in Consequences by Godley and Creme for the hell of it. Also SF Sorrow by The Pretty Things
[ninjaed]
Two of my favorites (that no one else has mentioned yet) are on the list.
Camel - “(Music inspired by) The Snow Goose”
Triumvirat - “Spartacus”
1984 by Eurythmics
Leviathan by Mastodon
Blood Mountain by Mastodon
Crack The Skye by Mastodon
Pink World by Planet P Project
Antichrist Superstar by Marilyn Manson
Warp Riders by The Sword
Red. White. Green. by Upon A Burning Body
The World Is Ours by Upon A Burning Body
The Audio Guide To Happiness Part 1 by Jolly
The Audio Guide To Happiness Part 2 by Jolly
Master Of Brutality by Church Of Misery
Pretty much everything by Insane Clown Posse including offshoots like Dark Lotus’s The Opaque Brotherhood.
ORO: Opus Primum by Ufomammut
ORO: Opus Alter by Ufomammut
Orogeny by Turbid North (this album is pure awesome)
Phaze 1 by Phaze 1
Pictures From Life’s Other Side by Johnny Dowd
Portal Of I by Ne Obliviscaris (another album that is pure awesome)
The Shape Of Punk To Come by Refused (this album is so far beyond awesome that I don’t think there’s a word that will adequately describe it)
11:11 by Rodrigo y Gabriela
I’m sure I missed a bunch that I own, but I’m at work and can only work with what is in my phone at the moment and what I can remember off the top of my head.
I deliberately left off soundtrack albums, with the exception of 1984 because while it was commissioned as a soundtrack for the film it ultimately wasn’t used in the final release.