I used to have severe insomnia, and would walk in the middle of the night, which would calm me down and relieve excess energy enough so that when I got home, I would go right to sleep. The police would drive through the neighborhood and we'd always wave at one another, but I live in an isolated neighborhood, so I would carry a knife and stun gun and sometimes mace with me, just in case. Around here, we have coyotes, bobcats (I've heard of cougars, too), and gators, as well as the occasional loose dog, so I figured I could handle pretty much any of those with what I carried. Many here do carry guns, but I don't like carrying heavy items when I walk.
I walk during daylight hours now, and mainly just carry a small blade with me. I keep it on my lanyard with my keys, but if I see someone acting odd, or I'm going through a section of woods, I will keep it in my hand, so I'll have better access in case I need it. I definitely recommend always being aware of your surroundings. I used to wear headphones when I walked on a local track, with many others nearby, but now I want to be as aware as possible, and want to hear if someone is approaching. I've also found that I prefer not listening to music when I walk, because I'd rather listen to the birds and other sounds, such as the fish jumping, since I walk on the local nature trails.