I do, mainly because the winds here are so drying. I use a couple sheets of wet newspaper, topped with grass clippings and leftover hay.The stuff I grow seems to like this combo, especially summer squash,peppers and cukes. We've got huge slugs, but they seem to like hiding under rocks more than under the mulch. If they are under the paper, they're not doing any damage that I can see. (Of course, now that I wrote that, I'll go out tomorrow and find they mowed down everything in sight!)
My big problem is figuring out when to put down the mulch. Too early & the ground doesn't warm up very fast. But if I wait too long, ragweed and crabgrass moves in. I try to wait until the plant stems are thick enough that cutworms won't go after them.
This year's experiment in frugal gardening is a red plastic tablecloth where the eggplant will go. Supposedly, tomatoes and their relatives grow better on red plastic. Don't know if this shade of red will work like the pricey stuff in the garden catalogues, but the tablecloth only cost 50 cents (an after-Christmas sale!) so it's worth a try.