Awards

Here´s a couple of awards that The Stones have had:

Grammy awards:

Best Rock Song for "Doom and Gloom"
Best Rock Album for "A Bigger Bang"

Best Pop Performance by a Duo or Group with Vocal for "Anybody Seen My Baby?"

Best Rock Album for "Bridges To Babylon"
Best Music Video, Short Form for "Love Is Strong"
Best Rock Album for "Voodoo Lounge" 

Juno Awards:

International Entertainer of the Year'

Mtv Video Music Awards:

 Best Group Video for "Love Is Strong"

Rock And Roll Hall Of Fame:

Performer award

 Sources: http://www.aceshowbiz.com/celebrity/the_rolling_stones/awards.html