That appears to be damage rather than something eating the leaves - looks as if someone or something has crashed through it, breaking leaves as they do.

It's happening to the leaves one by one. The plants are situated in an enclosed rear garden so no chance of people damage. The leaves will be "cut" off across the bottom, as in pic. or nipped off at the branch. Slowly but surely the branch becomes devoid of leaves. I have noted this same damage to other laurel bushes in garden centres. So stumped by this. I've tried all sorts of insect killing stuff but nothing seems to be stopping this happening. Anyone out there have any idea what it could be?

There are 3 common problems with this plant, which is Prunus laurocerasus, by the way, commonly known as Cherry Laurel, but the symptoms on display in your pics don't fit any of them. Maybe someone else will have an idea...

Thanks for your reply. Its puzzled me for sometime now. Maybe someone out there will have come across this. I live in hope.

Are the leaves missing outright, or are they showing up at the base of the bush? The leaves look torn, rather than cut, to me, Per. In the nursery trade, we call this sort of damage "truck blight"--for damage during shipping--or "finger blight"--for damage from customers thinking, "Is this a REAL leaf?" Is this a new plant, that you are just noticing damage that has been there since before you bought it? Or is it possible for children to gain access to the plant? Or maybe kids are dropping over the wall into that bush, as a dare or a shortcut?
Serious wind damage can also look like that, though the situation would be weird to produce that in an enclosed garden.