The boyband appear as the first all-male cover in the magazine's 40-year history.
Harry Styles said on his ladies' man status: "I feel so lucky to be in this position, I'm not entitled to get annoyed. I'm not half as busy as they make me out to be!
"A lot of the time, the way it's portrayed is that I only see women in a sexual way. But I grew up with just my mum and sister, so I respect women a lot."
On whether he had a type, he revealed: "There's someone I like... but this girl... isn't my 'type'. It's more about the person. How they act, their body language, if they can laugh at themselves.
"I find ambition really attractive too – if someone's good at something they love doing. I want someone who is driven."
Bandmate Zayn Malik said of their busy year: "Crazy! This award is a big honour. This year has been incredible, from winning a Brit Award to getting a number one album in America."
He admitted girlfriend Perrie Edwards from Little Mix has helped his confidence: "Yes, I'm a big believer in the motto, 'Behind every great man there's a great woman'. And she's great."
He continued: "I'm a couch potato. I love to stay in and just watch a DVD with the missus. Or we all go over to Louis's house and watch X Factor. Perrie is massively into it.
[Louis, Harry and Liam followed by fans at LAX Airport]
"That's what I look forward to most. I like going back home to Bradford and chilling with my mum and dad, and a cup of tea too!"
Of his girlfriend of a year, Louis Tomlinson said he doesn't doubt her around other boys: "She's gorgeous, but no, I don't. We both trust each other – I spend a lot of time out of the country on tour.
"She often flies out so I see her a lot. I think the key to a good relationship is being best friends too."
[Perrie meets Zayn's family]
Niall Horan was questioned about his confidence with girls and said: "I'm not really a flirt, I just try to be myself. I've got loads of good female friends, like Tulisa, but every time I'm pictured with her, people say we're going out.
"I don't really have a type, but for banter she's brilliant. She loves playing PlayStation. Tulisa is cool as f**k."
[Niall taking a stroll in LA]
On the coolest person he's kissed, he admitted: "Katy Perry. It was amazing and very purple – she had purple lipstick on. I don't think there will ever be anything cooler than kissing her… until I marry her maybe!"
Newly-single Liam Payne said of the other's love lives in the media: "Most of the time that's how I find things out! With Harry, everything happens so quickly. Harry is actually quite secretive.
"He tells us some things, but others he keeps secret."
The full interview appears in the December issue of British Cosmopolitan (on sale now). Behind-the-scenes footage can be viewed at cosmopolitan.co.uk/onedirection
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