How awful! It is of course Ground Elder and the roots go miles down, so you can't dig it out. Each tiny, minute, piece of root left in the ground makes a new plant, whatever you do don't let it set seed as it spreads that way too!
It's the bane of the gardeners life, I have a bed full of it and at last am going to resort to weed killer. I have dug it over regularly and picked up each 1/4" segment of root, but it does no good - it's rampant! As this is near a fence it may well have spread from your neighbour - have a peep over the fence and see if they have a bed of it in which case, it's not good.
See the RHS site here:
https://www.rhs.org.uk/advice/profile?PID=215
and here for how to tackle it:
http://www.gardenersworld.com/how-to/problems/weeds/ground-elder/469.html

Um, that's not strictly true, Honeysuckle - the roots don't go miles down at all, they're pretty near the surface, or at least no deeper than most other plants. The problem is, they'll regrow from tiny fragments of root left in the soil - the roots are quite fragile and break easily, and each root has tiny hairlike roots coming off it and the plant will regrow from those too.

It is possible to get rid of it, but if its next door too, both areas should be treated at the same time. How I did it was to spray the foliage around this time of year with glyphosate, wait a week or so and then dig it out - the glyphosate may make it look like the plant is dead, but its not, it regrows. What the glyphosate does do, though, is kill off those tiny hairlike roots so when you dig it up, you're more likely to get out all the roots. You will need to repeat the process with any regrowth - there will inevitably be bits of root you've missed when digging it up.

Thank you so much honeysuckle, and bamboo, I hate being beaten and am usually quite good at identifying plants.
We've just moved from Kent to Somerset, and had never seen it, we used to be plagued with horsetail and bindweed, none of that here but these new ones lords and ladies, and the ground elder.
I didn't realise I was spreading it ?thought I was getting on top,, ended up with a back ache to boot, it is next door too
So I'll have a word and see if we can get rid of it together. Thanks again.

Well, if its any comfort and I had to choose, I'd pick ground elder over horsetail and bindweed any day... I know ground elder can be got rid of, but that's certainly not true of the others.

Yor other question - yes it does flower but not spectacularly. It used to be grown in monastery gardens as a cure for gout..

And certain establishments sell a variegated version as an ornamental plant.

Or you could eat it.... https://scottishforestgarden.wordpress.com/2013/05/26/growing-and-eating-ground-elder/

Thank you all for your advice oh and steragram my husband sometimes suffers with gout so I'll get him to eat it ? job done lol
Glad I came on here very supportive cheers everyone

Thanks Bamboo for putting me right about how deep the roots go, but I have dug this (in places where the clay lets me!) to a depth of 1 ft and was still breaking off bits of root that were deeper. In other places it's only about 6" down which makes it easier. Difficult to get rid of when it's growing around the bottom of a hedge you can't dig up and amongst plants, but will try your glyphosate treatment.

Worse in clay soil, Honeysuckle - I recall it took me three goes to clear it, so around 18 months. Persistence, as always, is key, I was determined to prove I was more persistent than the blasted ground elder...

Lords and Ladies are left where they grow in my garden. I like them. Should I be getting rid?

Not necessarily, if you like them - the berries are highly toxic, but anyone or anything that tries eating them spits them out quickly, feels like eating broken glass or having a mouthful of needles, or so I read...

Alex your husband will be cursing me - I don't think it tastes very nice although I believe it was used as a pot herb at one time.