Female journalists say that Peaches looks like a cross between Lolita and Marianne Faithfull; male journalists are not allowed to say this sort of thing. In reality, Peaches looks like a 17-year old wearing a lot of eye-liner . . . and it doesn't get much sexier than that.

One of the best things about Peaches is her shambolic, slutty style, which is four parts Top Shop and Miss Selfridge to one part Missoni. She came in at number seven on Tatler's Fashion Icon list, and was the youngest of the group by at least 15 years. Tatler's citation reads: "Has the scruffy, teenage King's Road thing down pat in cut-off layers. Sexy rock chic.

Watch her style evolve."

Meanwhile Peaches has become famous for what are imagined to be major arguments with her father, Sir Bob. After all, she is a teenager.

She has had her nose, tongue, navel and ears pierced and has two tattoos.

Out late one night, she shouted at photographers:

"Don't take my photograph, I'm not supposed to be here!" Rows are said to be about her pocket money . . . Peaches has allegedly said that she's not getting enough, although her own earnings surely make this unlikely . . . and her boyfriends. She once dated Llewelyn Graves, grandson of the poet Robert Graves. Now she is said to be seeing Donny Tourette of the rock group Towers of London. The Towers of London motto is Drink, Fight, F**k. It is an eerie echo of Bob Geldof 's old remark that he went into rock to "get rich, get famous and get laid".

At the same time, Bob has stood up for Peaches, saying that she is "just a normal child" and "an alright kid". Peaches has said that she approves of the idea of curfews and of parental discipline. She often sounds like the public Sir Bob, although she is far more fond of spending money.

She advises teenagers to "watch the news". She has allegedly said (there is a lot alleged about Peaches, because so much of what she is quoted as saying comes from tabloid celebrity columns) that she is unhappy with her looks, which she thinks are too much like a feminised version of her father. She is, however, slightly under five foot five inches tall.

Peaches has become such a byword for naughtiness that, when the shoplifting controversy broke out, old ladies were heard to mutter over their copies of the Daily Mail: "It's always Peaches."

Peaches was discovered leaving the Oxford Street branch of Urban Outfitters in a coat which the store claimed she had not paid for. The security tag was still attached to the coat, which was priced at £235. An hour-and-a-half later, a rather mortified Peaches departed Urban Outfitters by the back door, stepping in to a silver Mercedes which had been sent to rescue her. The photographers bayed at the front. Peaches later strongly denied shoplifting.

The only sensible response to the incident came from a member of the public who wrote in to the Daily Mirror website to say: "If her stupid father had not given her such a ridiculous name she would not have to suffer so much media attention." Peaches herself has said: "I hate ridiculous names. My name has haunted me all my life."

It seems strange that such a decisive girl has not dumped the name; perhaps she keeps it because it was given to her, not by her secular saint of a father, but by her flamboyant and much-maligned mother, the late Paula Yates, who died when Peaches was 11 and whose public persona she so closely shadows.

Peaches is famous for being young and famous. She and her younger sister Pixie turn up at film premieres and rock concerts as the answer to any photographer's prayers. The two of them already have their own fan website, which is run by a young woman called Hannah. On the website, you can read some of Peaches's words of wisdom, which have included: "Jane Austen and the Brontes: boring feminist crap"; "Stop being a consumer: fight the power"; and "It's like teenagers today have nothing to rebel against." And on Tony Blair: "It's hard to run a country and I think everyone forgets that."

Her older sister, Fifi Trixibelle (23) works as a receptionist in a health club.

Pixie, just a year younger than Peaches, seems a sunny young girl and has a surprisingly deep voice, a sort of horsey rumble. Both she and Peaches talk like the rich Chelsea girls that they are.

But, although Pixie is the beauty of the family and poses for photographs and attends premieres, only Peaches, it seems, wants to be a part of the media.

She set out her stall when she attacked Trinny and Susannah, of What Not To Wear fame, as "upper-class bitches with no fashion sense". Trinny and Susannah regally responded that they too had been young once.

Peaches has expressed her wish to go to university in New York to study English and journalism. She is already a nice writer and has had columns in the Daily Telegraph and the teenage version of Elle. Here she is writing about her programme Teen America, made for Sky television. She had witnessed Born Again teens swearing to abstain from pre-marital sex: "The pledge was 85 words with little punctuation, and would leave most teenagers too breathless to even contemplate coition." Even allowing for the talent of the Daily Telegraph sub-editors, this is neat work, and a funny idea.

Like her mother, Peaches is a straightA student. Her television work, including a recent report on body fascism within the fashion industry, must be done on her half-terms and holidays.

The young singer Lily Allen hated Peaches presenting A Beginner's Guide to Islam, on Channel 4. Lily Allen said she wanted to kick Peaches after watching the programme. But Peaches emerged from it with great charm and cleverness. Sent to live among strict Muslims in Morocco, her teenage hostess grew to love her, and to admire her fearlessness.

Peaches gamely gave up her fashionable clothes and make-up, although she got cranky about the early starts and had problems as a vegetarian in a country where the greatest delicacy is goat.

She handled the numerous sexual advances of the men with humour and sophistication. Best of all, she asked some interesting questions. Meeting a young man who lives in a cell reciting the Koran all day, she asked him if he never thought of a lovely Fatima to keep him company. No, said the young man, never.

Peaches didn't push the point; she didn't have to. The young scholar was immediately exposed as a hypocrite who was afraid to tell the truth.

In her wit and her cleverness and her happiness in front of the camera, Peaches is very reminiscent of her mother.

Paula Yates has somehow been written out of media history, but she was a ground-breaking television presenter in her day, and has never received the credit she deserved as the first TV female who combined light-heartedness with being smart in both senses of the term.

Peaches is young enough to drop from the public gaze altogether, if she ever wants to. She has had so much public attention so young that she may tire of it. In the meantime, we should enjoy her zest while we can.

C.V.

Full name: Peaches Honeyblossom Charlotte Angel Vanessa Geldof
Born: 16 March 1989
Lives: Chelsea, London
Occupation: Schoolgirl
Hobbies: Broadcasting and journalism; DJs with another girl as the Trash Pussies; Is a vegetarian
In the news because: It is genetic. Recently she's been accused of shoplifting, which she strongly denies. Is also in an alleged row with her father Bob about her boyfriend