What does it feel like to get punched or kicked in the face by bare knuckles or feet?

It depends on where in the face, and it depends on the circumstances.

In one school where I trained, we used to spar with "nearly" bare knuckles -- wearing light (and by light I mean "cheap") football handpads with no more than 1/8" thick padding. The idea was to mitigate the chances of getting cut by a punch while still allowing us to feel what it might be like to give and take a real punch. The first time we sparred, getting punched didn't hurt much, mostly due to the adrenaline. Over time, though as full-contact sparring lost its novelty -- and thus, its adrenaline -- some of those punches came to hurt. In particular, punches to the nose (even glancing ones) and eye area tended to hurt right away.

Also, you "hear" a certain sound when you get hit. It's a somewhat unique, dull thud -- I'm sure one's perception of sound is affected by the fact one's head is being impacted, but I'm not sure how. All I know is, nothing else sounds like it.

In real fights, the adrenaline protects you from the pain. A really good shot might disorient you for a fraction of a second (I've never gotten knocked out). But you don't really feel much discomfort.

In all cases, regardless of whether you feel it at the time, if you take one or more punches to the face, you're going to feel it the next morning. Standard practice among my sparring partners and me was to automatically take two Advil (or your analgesic of choice) before bed that night, regardless of whether it hurt or not.
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