Any chance of a picture? Have you watered it sufficiently? Have you given it different conditions from those it was previously growing in?

Thankyou for your response...
I think I may have watered it too much, rather than insufficiently.
I don't have a photo, I'm afraid...not very good with getting pictures onto the computer.
However, I can tell you about it...
It's branches are trailing, about 9 inches long and it came in a plastic pot.
When I got it home I watered it thoroughly, then realized that I probably ought not to have done.
So I changed its pot for a terracotta one, in the hope that its compost would dry out a bit and become less waterlogged.
I live in Southern Spain and the Lavender plants I bought at the same time are very happy in their terracotta pots.
But the leaves of my poor little Trailing Rosemary are, now, beginning to fall off, leaving the end of the branches bare.
There are still (a very few) green leaves, right at the base of the branches.
Would it be worth persevering, or do you think I've killed it?

Rosemary are a challenge to grow. Once they are sick it is usually a terminal situation. Ideally, rosemary should be in full sun and not under or over watered.