My run is covered with chicken wire with 1/2 of it having a silver tarp over it to keep the rain & snow off. It works really well as long as I remember to make the rain drain off and keep the snow from getting too deep on it. The first year I had it the snow got so deep and heavy it broke the framing.
My turkey run is covered with aviary netting draped over arched pvc piping this is only to keep my turkeys from flying over the fence. My chicken house is covered with chicken wire to stop hawk attacks(the lower portion of run is hardware cloth)I do have a tarp I tried to secure over the run but after numerous rain storms and water eing caught in the tarp and it started making the wire of the roof stretch and sink down. It was a pain so I took the tarp off. My newest chicken coop I decided to try a fully roof covered run with roof framing and plywood and it is shingled. I love it. It looks nice keeps the chickens dry an pretty much mud free Im so glad I did it and am in the process of trying to come up with ides to give the other coops roofed runs.
Mine is actually quite strange! Since the only thing we have to worry about is hawks (there is nothing to climb into the run during the day, that we know of!) we simply have neon twine stretched out across the run. This allows full sun and air in, but would stop any dive-bombing hawk, as its wings would get tangled in the twine. This also keeps my NHR hen from jumping up onto the fence or coop! We have not finished installing it completely.
Oh and my coop is a wooden one i designed and built. I painted it with white outdoor paint. to hopefully keep it pretty cool inside, since i live down in southern louisiana.
The roof on it is a sloped wooden one, and it lifts in two sections to allow for easy cleaning. I am adding on to the back of the coop, and that will have a similar lid that will open for easy cleaning and egg collecting.
Part of my run is covered with the roof from the coop. I ended up adding on a larger run that was initially left uncovered. After losing a chicken to the heat last summer, the run "annex" is now about 1/3 - 1/2 covered with shade cloth, the rest uncovered. No problems in 2 1/2 years with predators - but I do have a fenced yard.