This might sound like a cop out, but it depends on what the weapon is. Specific dedicated throwing weapons did exist. Off hand the Javelin comes to mind (I know, it’s not what you’re thinking of, but there it is). Dedicated throwing knives do exist. I’m not sure if there were dedicated throwing axe designs, or if it was just re-purposing convenient hand axes.
Getting hit with a javelin is bad news. Some varieties, like the Roman Pilum, were designed to penetrate shields. The Pilum was also reportedly able to punch through armor, due to its design. It also sidestepped the real problem with most thrown weapons: Even if you kill someone in the process, you’re giving your enemies an extra weapon to use on you.
Getting hit with a flying axe is not going to do you any favors. Axe strikes are nasty, and having one lobbed into you is close enough.
It’s one of those weird cases where, I know it happened, but I don’t know much about it’s actual use in warfare.
Modern axe throwing is usually end over end. With the axe spinning on the point of balance. You can find videos of this pretty easily. As with knife throwing, it’s a hobby for some people, and you can even buy targets marketed for throwing axes. The modern throwing axes I’ve seen favor a single large curved blade on the head, and a haft that curves towards the blade. I have no idea if that’s just ergonomics, or if it actually helps with throwing. I suspect the latter, but, that’s just a guess.
The one thing I’m fairly sure of is you weren’t throwing your only axe. You’d throw spares, so you still had a weapon on you. The last thing you want to do is give someone else your last weapon.
Throwing knives are really more of a party trick than a viable combat option. It’s one of these things where films and media makes it look a lot more useful than it actually is. In the handful of situations where you just need to put a knife in that guy over there, and walking 15 feet sounds too much like work, throwing a knife is a legitimate option. Otherwise, you’re better off delivering it by hand. Into their kidney or through their favorite artery.
Or you could shoot them. There’s no risk of them throwing the bullet back. Just a thought.
Dedicated throwing daggers tend to be just a blade without much, or any grip, since a knife is usually thrown by gripping the tip between the thumb and forefinger, and flicking the wrist while releasing the blade. I’ve never been able to get the hang of it, though I do have a real knack for misjudging the spin and connecting with the target with the handle.
Throwing swords is a very bad idea. It did happen. There’s actually surviving training manuals that talk about it, and suggest methods. Those same training manuals will also call you an idiot, well, technically a “knave,” for even considering it, but, it can be done.
Historically, a thrown sword would be gripped over the shoulder, across the guard, fingers toward the pommel, palm on the flat of the blade, and then thrown in line with the blade, like a spear.
I want to say actual use of this was mostly restricted to judicial duels, but, I’m not completely certain. Judicial duels would also see combatants bringing swords with threaded pommels, unscrewing them before the duel and then throwing it as a distraction. There’s a lot of question as to how, exactly, that worked, but it does pop up in training manuals as something to do or worry about your opponent doing.