I use a LOT of grass clippings from the lawn on my veg garden... I put it on in relatively thin layers so it dries out, but by the end of the summer have a pretty good amount everywhere. It helps keep the moisture from evaporating, keeps the soil from splashing up on the plants when it rains, suppresses the weeds (the ones that do grow are a lot easier to pull), and is much cleaner for walking around in the garden. Plus, I've added a great deal of organic material to my garden over the years by doing this, making the worms really happy and vastly improving the drainage of the soil.
In my ornamental beds, even the shady ones, I use a shredded white cedar mulch (which is produced locally) -- I like the way it looks, and again it does a great job of suppressing the weeds. Which is definitely not to say it eliminates weeds... but it helps!
