It looks like Convolvulus, Alex. It is a native wild plant that would normally be considered a weed in the garden. However, it will either have large, white trumpet flowers or smaller, pink and white flowers. It does look lovely in hedgerows but possibly overpowering in the garden.
Wait for further id from other Goyers, though, before you yank it out.

Bindweed, as far as I'm concerned, that's all you need, I consider it a reason not to move into a house with a garden that has it. Get it out while you still can.

If the flowers open into small trumpet shaped white or pink white flowers, then it is bindweed, but although the leaves look right, it could be another type of convolvulus or ipomoea.
Don't let Bamboo scare you! I know he's joking, but I've never lived anywhere without a good bit of bindweed, and it isn't that hard to manage if you're determined.

This is actually Black Bindweed (Fallopia convolvulus) and is an annual climber, not the true Bindweed (Calystegia) so not as bad as you first thought.