This is my rule of thumb. Netting will keep out other birds. A determined bird could chew through it if they wanted to. Most raptor-birds-of-prey do snatch-and-grabs and therefore it will keep flying predators out and flying chickens in.
Chicken wire will keep your chickens in and some dogs and smaller predators out. A determined dog can chew through chicken wire (sounds painful). A raccoon can def. chew through chicken wire. Not so sure about opossums but they look nasty enough to do the job. Same with foxes and coyotes.
Hardware cloth keeps everybody out. For extra security put hardware cloth on top of the soil (grass, gravel, etc.) around the parameter from the bottom of the run/coop extending out 2 feet and stake it down with landscaping pins or clips.
Apparently, animals that dig like coons & dogs aren't smart enough to back up two feet and start digging there. So when they dig next to the coop or run they soon run into your wire and give up.
I dont have much problems with predators so the top of our run is chicken wire and the sides are hardware cloth with the aforementioned ground skirting of hardware cloth. Coop gets "locked" (closed) at night. If you lived in an area with alot of those predators you may need to be more sturdy.
Also, during the day the dangers are flying predators and dogs (cats if you have chicks or small breeds). At night is when you shoud really worry - Owls, dogs, cats, coons, possums, coyotes, foxes, skunks and even snakes.