In shade, Ajuga reptans, Geranium maccorhizum, Lamium maculatum varieties, Campanula muralis - the latter is fine in full sun as well, as are Iberis sempervirens, Arabis, Silene uniflora, Hebe pinguifolia, Hebe 'youngii', and Helianthemum.

I like creeping Phlox, though the flowers are over all too quickly. Dianthus species like pinks are good and come back into flower if you dead head them. Arabis and Iberis, Aubretia.
Oh, just seen Bamboo has covered some of those already....

All that said certain weeds can grow though anything!

There are several attractive low growing creeping thymes.

There are so many perennials that will fit this bill.

Think outside the box and try a dwarf alstroemeria - choose on that's only 6 or 7" high and it will spread all over that bed/border giving you flowers from early spring to the first frosts, non-stop, they are incredible value for money plants.

I would offer a vote for Lamium maculatum 'White Nancy' lovely white leaves with green edges (certainly more white than green) - grows in most positions and has little white flowers aswell. I put a tiny plant in in May (£1 from Wilkos) and its already a couple of feet wide but barely a few inches high.

I'd echo that, Weatherwise - I do mention Lamium maculatum varieties, but White Nancy is definitely one of my favourites, although in my garden, it has lilacy pink flowers... and its definitely White Nancy.

I'm sure you're right Bamboo - mine isn't flowering yet but it was teeny when I got it. Grown so quickly.

Actually, I'd be grateful if you could let me know whether yours has white or pink flowers, Weatherwise - I haven't got time to check, but I remember being surprised when mine flowered pink, because I was expecting white flowers, so if yours is actually white, then there's obviously two very similar varieties knocking around with the same name - or mislabelled!

Will let you know.

Hi Bamboo. I said I'd get back to you - my White Nancy has white flowers. I'll put a pic up. For info Silver Beacon Lamium has similiar leaves but pink/purple flowers (http://www.youcanlearnseries.com/Landscape/Plants/Lamium.aspx)

I'm mystified, Weatherwise - the one I've got planted definitely isn't Silver Beacon, or at least, the Silver Beacon I have elswhere doesn't have the large creamy white area on the leaves, but this plant definitely has lilacy flowers - and was sold as White Nancy. And by the way, it's now taking over the garden - its spread into an area 1 foot wide by 6 feet long, I'm constantly hacking bits off.