I've seen either 2 or 3 black bears while running. I saw the first one, an adult, here in Utah a few years back. It was at night and kind of alarming. The thing was on the middle of the dirt path going down Collins Gulch at Alta Ski Resort, and I had no other way down, so I had to wait for it to move to the side and then run past it. I went by slowly, but once I was a tenth of a mile away, I hauled it the rest of the way down the mountain. Last summer, I saw a cub at mile 30-ish of the Old Dominion 100 mile run. Roughly 15 miles later, while coming back near the same point on the course, I saw either the same cub again, or another one of about the same size. The first time, it sprinted across the road about 50 feet in front of me, but the 2nd time, it was only 5-10 feet in front of me when it crossed the road. I never saw the mom, and I was definitely a little nervous about running into her. In any case, black bears are probably not usually that big of a deal. If you see a grizzly, you might be in trouble, but not usually with a black bear.
Regarding moose, I've seen somewhere in the ballpark of 15 of them out here in Utah, mostly in the Cottonwood Canyons, but 1 in Mill Creek Canyon, and 1 in Neff's Canyon. As long as they don't have young with them, they're not usually aggressive. Just give them some distance and they're cool.