Asked by Dick Cavett, in one of the old clips scattered about this high-powered concert film featuring the Stones, if he could picture himself at the age of 60 doing what he did then, Mick Jagger replies: “Easily, yeah.”

One doubts, however, that he could have imagined performing in a film directed by Martin Scorsese, with half-a-dozen Oscar-winning cinematographers in attendance at a birthday concert for Bill Clinton and a benefit night for the ex-American president’s foundation. How we love our gnarled old rockers these days!

But hold on. Is this really the most appropriate setting for the veteran rebels to perform in? It’s 2006 and we are at the small Beacon Theatre in New York, packed with an audience of finance high-rollers and hedge-fund yuppies. There is not a sniff of danger — musical or otherwise — about the proceedings.

The still wafer-thin Jagger, jigging energetically about the stage, and his band of heavily sweating co-celebrities, are accompanied by a fair number of supporting musicians. Among those helping Mick out are Christina Aguilera, who wasn’t even born when he wrote Live With Me, the song they perform together. There’s also Buddy Guy when they get to Muddy Waters’s Champagne and Reefer, and the minuscule Jack White.

The extras give the seniors a bit of breathing space, but on their own they can still work up some magic — even a tired-looking Ronnie Wood as he pedals his way through Far Away Eyes, and especially the inimitable Keith Richards when the time comes for his solo. He introduces it with a time-honoured joke that still raises a laugh, even from this stiffer than usual audience: “Hello, it’s good to see you all. Actually, it’s good to see anybody.”

It is abundantly clear, though, that Jagger is the heart and soul of the outfit, the man who sings and struts the most and, as the film shows, organises everything — whether Scorsese likes it or not.

Mick must, like the rest of the Stones, have more money than he knows what to do with yet his mental enthusiasm is still palpably there and so, obviously, is the purely physical fitness.

Some time before the two-hour-long Shine a Light finishes, you begin to wonder exactly how much input Scorsese has had — he doesn’t even appear to know the running order of the numbers at one point. He has 18 cameramen at his disposal (including the redoubtable Albert Maysles, still toting a hand-held job at his advanced age) and an editor, David Tedeschi, who spent nine long months piecing together the footage from two separate concerts.

When you see the director at the very end shouting at his crew to raise their cameras skyward, away from the landmark theatre and up into the heavens, you wonder if this is finally the point at which he determines to get his way without Jagger interfering.

Shine a Light is all done with ace professionalism, both on and off the stage, but there is no sign of the Scorsese trademark that was evident in his The Last Waltz (with The Band) and No Direction Home, his perceptive documentary about Bob Dylan. He worked on Woodstock too, of course.

But Stones fans won’t bother about the little matter of directorial control. All they’ll want to know is that if you want to look up the group’s noses, you’ll be able to do so. This is, above all else, as intimate a portrait of the aged band as could possibly be imagined. It’s warts and all stuff, and it sometimes feels as if you are right there on stage with the self-styled greatest rock band in the world.

They don’t give us short measure, though some of the early clips are interspersed haphazardly and there is no mention of one Brian Jones (whom I would have much rather seen than William Rees-Mogg and that daft Bishop asking silly questions about morality and rock). You do get to see the great Bill C and, very briefly, his wife and her mother. We don’t really know whether they stayed for the music. But if they did, perhaps they wore ear-muffs. Some of you, unused to this proximity, may feel that you need them.