My favourite houseplant but what on earth is it? Help!

I need some help identifying my favourite houseplant. I've posted a few pics on Flickr, please follow links below. I got the plant from my mother in law who picked it up from a craft market in Somerset. It is my favourite houseplant of all time and friends and guests always comment on it when they see it my house--I've tried Googling it and tracking it down to no avail! Can you help me with the name of it?

The plant:
http://www.flickr.com/photos/quailbymail/3887048648/

The leaves:
http://www.flickr.com/photos/quailbymail/3887049232/in/photostream/

The base and babies:
http://www.flickr.com/photos/quailbymail/3887048952/in/photostream/

The plant has smooth fleshy leaves, likes to turn it's leaves to the sun, fast growing, likes water and a bit of drought, creates lots of little babies and incredibly easy to care for. Any have any ideas?

Thanks a lot!!!

 

Hi, perhaps a Peperomia? They are quite small, around 30cm max. Here's a link to a photo.
http://www.desert-tropicals.com/Plants/Piperaceae/Peperomia_polybotria.html

 

You know what? I think that's the closest thing to it and where I'll start my research!!! Thanks a lot!!!

 

Your plant is Pilea Peperomiodes, very peperomia like.