Nirvana - The Man who sold the World. (Bowie)
Faith No More - Easy (Lionel Richie or what was the name of his group?)
Jeff Buckley - Hallelujah (Leonard Cohen)
Sinead O'Connor - Streets of London (?)
Jimi Hendrix - All along the Watchtower
Wet Wet Wet - Love is all Around (The Troggs)
Pixies - Head On (Jesus and Mary Chain)

...can´t remember more at the moment.

Do you have any?

H2O - Like A Prayer (Madonna)

My all-time favorite cover song, and I don't even like the original.

Quoted from opportunityknocks

Faith No More - Easy (Lionel Richie or what was the name of his group?) = The Commodores
Sinead O'Connor - Streets of London (?) = Ralph McTell

'Someday' by Shawn Colvin (Steve Earle original)
'Cause We've Ended As Lovers' Jeff Beck/Eric Clapton (Stevie Wonder)

Rammstein - Stripped (Depeche Mode)
Gary Jules - Mad World (Tears For Fears)
The Beatles - Twist and Shout (Can´t remember the name)
The White Stripes - I Just Don't Know What To Do With Myself (Dusty Springfield)

people may disagree with this one with i prefer in from the original

Van Halen-You Really Got Me (Kinks)

I'm losing you (The Temptations) covered by Rod Stewart
Behind blue eyes (The Who) covered by Limp Bizcuit. Can't say it's better, but I like it.
California Dreaming (The Mama's and The Papa's) covered by The Beach boys
Bad Boy (Larry Williams) covered by The Beatles
And more to list as I think of 'em

By the way Oppotunity knocks, the band you could'nt think of was The Isely Brothers.
Which brings another to mind.
This old heart of mine(The Isely Brothers) covered by Rod Stewart. Rod has a knack for the old Motown stuff

'Dreams I'll Never See' by Molly Hatchett (originally done by Fleetwood Mac)
'Bang A Gong' by Powerstation (originally done by T.Rex)
'Spirits In The Night' by Manfred Mann (originally done by Springsteen)
'It Aint Me Babe' by the Turtles (originally done by Bob Dylan)
‘Mr Tambourine Man’ The Byrds (originally done by Bob Dylan)
'You Keep Me Hanging On' Vanilla Fudge (originally done by the Supremes)
'Aint That A Shame' Cheap Trick (originally done by Fats Domino)

more to come later

Gunes and roses-Sympathy for the devil, originally done by the Rolling stones.

Aretha Franklin- Respect (originally done by Otis Redding I believe) is one I don't think anyone mentioned

Byrds- All I Really Want To Do  (Bob Dylan)
Frankly Mr Shankly - High Llamas (The Smiths)
Shipbuilding - Suede (Elvis Costello)

Me and Bobby McGee

who orrigionally did that?

^ Kris Kristopherson wrote and recorded "Me and Bobby Mcgee" And then Janis Joplin and a multitude of others covered it.

Quoted from Wordno

Gunes and roses-Sympathy for the devil, originally done by the Rolling stones.

That version is no where near as good as the rolling stones version

I like Van Halen's cover of Pretty Woman better than the original.

I personally thought Eric Claptons version of knockin on heavens door was better than bob dylan but thats just my opinion

(dont fear) the reaper - ceasers.. that damn cowbell is finally gone
also born under a bad sign by cream but ive never really heard the original

I'm probably going to get lynched for this, but I think Letters to Cleo's cover of I Want You to Want Me is better than the original.

Also, Chantal Kreviazuk's cover of Leavin' on a Jetplane is the best recording of that song ever.

"AS GOOD, BUT NOT NECESSARILY  BETTER..."

The Sundays: "Wild Horses" (wonderfully ethereal version of ultra-great original by the Rolling Stones)
Santa Esmerelda: "Don't Let Me Be Misunderstood" (orig. Anilmals)
Stevie Wonder: "We Can Work It Out" (orig. by some Liverpool club band)
Stevie Wonder: "Blowin' In The Wind" (orig. Bob Dylan; I also like this version better than Peter, Paul, and Mary's folkie hit cover)
Elton John: "Lucy In The Sky With Diamonds" (orig. by mid-1960's psychedelic band)
Anne Murray: "You Won't See Me"  (orig. by mid-1960's folk/rock group)
Chicago: "I'm A Man" (orig. Spencer Davis Group, really Steve Winwood)
Blood, Sweat, And Tears: "Got To Get You Into My Life" (orig. by mid-1960's white soul group)
Smith: "Baby, It's You" (far better than original by Shirelles or Beatles' cover)
John Lennon: "Stand By Me" (great orig. by Ben E. King)
Paul McCartney: "Don't Get Around Much Anymore" (pre-rock classic written by Duke Ellington)
Olivia Newton-John: "If Not For You" (orig. Bob Dylan; my most controversial pick; I also like it better than ALL THINGS MUST PASS version or other Dylan/Harrison combined version from BOOTLEG SERIES)
Olivia Newton-John: "The Air That I Breathe" (written by Albert Hammond; first released by Phil Everly; ONJ's version is even better than great hit version by the Hollies)
George Benson: “Love Ballad” (one year after inferior orig.by LTD/Jeffrey Osborne)
George Benson: "Here Comes The Sun" (not as good as ABBEY ROAD original, but better than Richie Havens' hit cover)
Blind Faith: "Well All Right" (orig, by the great Buddy Holly; Santana later copied this cover)
The Beatles: "Twist And Shout" (actual original by the Top-Notes; hit single by Isley Brothers)
The Beatles: "Long Tall Sally" (orig. by Little Richard)
Carpenters: “Please Mr. Postman” (I like this better than original by Marvelletes or Beatles' cover)
Buddy Holly: “Love Is Strange” (posthumous version repeated and overdubbed with strings; orig. by Mickey and Sylvia)
Nilsson: “Without You” (I love Badfinger, but this is the biggest improvement on an original I have ever heard; produced by the great Richard Perry, my second favorite producer after George Martin)
Anne Murray: “Danny’s Song” (original by Kenny Loggins)

And of course, both Gladys Knight's (my favorite) and Marvin Gaye's versions of "I Heard It Through The Grapevine" are, technically, covers of Smokey Robinson's surprisingly forgettable first version.  (There was also another interim Motown recording by Undisputed Truth that I have never heard.  And I get nothing from CCR's version.)

(I also require that the cover version be somehow different than the original version, not just the same arrangment but by a better singer or demonstrating a higher fidelity production, as with Linda Ronstadt's ones, which I greatly enjoy but usually just followed the familiar recording.)

almost every elvis song

Crossroads- Cream

I´ve never heard Buddy Holly´s version of Words of Love? Is it good?

Not fade Away-- stones

orignaly done by Buddy Holly

I agree with the Stone's version of Not Fade Away, but I like Holly's too. If no one mentioned this before, and I don't know how many would agree, but I think Nirvana's cover of The Man Who Sold the World is better than Bowie's version. I love it.

Instead of opening a new thread I decided to use this one - I've just heard the Pearl Jam cover of the Who's "Love, Reign O'er Me".
Personally, I like the cover - Eddie's vocals are just amazing but I still think Roger makes it better. I don't know why, maybe because I feel and hear much more in the original - something typically from that time, something very powerful, something everlasting. PJ's version is sweeping in the way a muddy river runs through a forest after heavy rains but it's emotional only in one direction, like, for example, pain. full stop. or great happiness. only this.
I don't think I expressed well what i wanted to but I'll be glad to hear what you think:

Pearl Jam's cover

Original song

If you want cover versions RUSHs FEEDBACK cd is brilliant its a tribute to the bands RUSH were influenced by .

Summertime blues
Heart full of soul
For whats its worth
The seeker
Mr soul
Seven is seven
Shapes of things
Crossroads

These are as just as good as the origanals, Alex Lifeson has never sounded better!  

Maybe it's not better than the original, but I have been listening to Dusty Springfield's version of Piece of My Heart lately and it is just as good as Janis' version. She puts a whole different spin on it and that's what makes it good.

You Send Me- Steve Miller Band

i honestly like it more than Sam Cookes version, even though Sam is one of the greatest vocalist in history

Quoted from mr kite

If you want cover versions RUSHs FEEDBACK cd is brilliant its a tribute to the bands RUSH were influenced by .

Summertime blues
Heart full of soul
For whats its worth
The seeker
Mr soul
Seven is seven
Shapes of things
Crossroads

These are as just as good as the origanals, Alex Lifeson has never sounded better!  

Heard some of these before. Rush is a Brilliant band!!!!!!

Ever hear Jane's Addiction's version of  Sympathy for the devil???????? TOO FAST!!!!!!!!!!

Speaking of Stones covers, I am totally in love with Tina Turner's version of Honky Tonk Woman. Maybe not better than the original (not as much cowbell) but she certainly puts a lot of attitude into it.

Anyone ever heard Snoop Dogg's "Bitches Ain't mess" done by Ben Folds? VERY funny, and in his version it's actually a very good song.

Something in the Air, The Heartbreakers version is the best.