Waterproof gloves/mittens

I have Sealskinz gloves for the bike.  I wouldn't use them for running though - I have a pair of thin fleece gloves, which are not waterproof (obviously!), but they don't hold the water, so my hands don't get cold.

My hands never get cold running, come to think of it.  Even in very cold weather the gloves tend to get taken off after a few miles as my hands get hot.

I've just discovered Decathlon hand warmers (foot warmers also on offer). Small sachets that warm up on contact with air and stay warm for 5+ hours! Decathlon ski socks, £8.99, are a bit thick but very warm and knee high so fab for those very cold days. Not sure if pros would approve but comfort more important to me!

i have some split mittens fro Craft, on a bike they keep my hands warm even in sub zero conditions, ive ran with them in -15 and they worked well, but i found anything over zero i got too hot in them.