You definitely can't have a charcoal grill, because they emit too much smoke and they can be a fire hazard. My condo has a large deck but there's no overhang cover on it. So we've got a pretty large gas grill with 4 burners.
When I moved into my last apartment, I was assured by the leasing agent that a grill on the patio was OK. Then the first spring came around and notices started going out that grills had to be at least 20 feet away from the base of the building. Of course, I ignored the notices and never got in trouble, but I've heard of apartment complexes calling the fire dept on people who do use grills on their patio..
Yeah, it's been steadily appearing as notices and then as leasing terms over the past couple years. My lease terms say no closer than 15 feet.