Thanks to Greek amphorae, we have a large number of images showing Greek and Hellenistic life (other areas of the Mediterranean influenced by Greece). This little picture illustrates the all-in form of Greek wrestling called pankration. Pankration never went away and never lost favor and today, it lives on in the form of mixed martial arts. There are certain pankration moves that are forbidden in competition, but they definitely wouldn't have been left out of any real combat.
There's no doubt that the topics of one of my favorite shows, Deadliest Warrior, cause renewed rivalry and competition between advocates of history's greatest warriors. Over on the Hungarian blogs, I'm pretty sure they're still crowing that Attila the Hun beat Alexander the Great on the show.
Wikipedia has some dedicated chroniclers, so you can see the breakdown of weapon-to-weapon simulation results that produces "winner" and "loser" on the show. [Wiki note: they are slowly "winning me over" - these Deadliest Warrior entries are great, and their year-by-year entries are brilliant, so I guess this now trumps Wiki-sexism and Wiki-promotion of nonauthors with bigger and more comprehensive entries than Nobel and Pulitzer prize-winners].
Here's my take - and my best friend is Hungarian so I know Attila is Leader of the People, and so far as I know, I have no Greek ancestry.
Alexander should have won. From all that I know, while Attila was a great military leader and conquerer in his environment and time, Alexander had qualities of military leadership that I think would make him a great leader and conqueror in any time. The Wiki weapons breakdown chart shows that the deal-winner in Attila's case was his strong superiority in the use of the Hunnic bow on horseback (the real battle-winner for the Huns in real-life, too) against Alexander's use of the gastraphetes, or giant crossbow, that had to be braced against the midsection while standing securely on-foot. The weapon-by-weapon chart results show the occasional fundamental weakness of the four-weapon matchup system of Deadliest Warrior. First, weapons have to be selected and then relatively evenly-matched. And in this case, these two weapons were used for different purposes and in different ways - the matchup isn't as valid as sword against sword, or bows that were both used on horseback.
It's possible that Attila had a hand-to-hand martial arts training that has been overshadowed by the Hunnic bow and horseback method of Hun conquest over time, which meant that he could match up with Alexander in a pankratic wrestling match. However, matching Attila's battles with Alexander's shows equal ruthlessness, but also a clear strategic advantage to Alexander. I think that Alexander's known battle history illustrates amazing strategic and tactical thinking, while Attila seems to have been a more straight-ahead fighter doing a different sort of battle and conquering much more disorganized enemies.
We can't know what really would have happened, of course, and of course we do not really know these two men, we only know the marks they left on history and the world. It's possible that Alexander had an "Achilles' heel" in that he was known to party to excess (a trait that eventually cost him his life at an early age), and that he might have had some illness that affected his mind - because he definitely started to believe the "I am the son of Zeus" stuff there at the end. There are no such suggestions with Attila - he was a hard-bitten warrior and conqueror from start fo finish.
I still think that in a slightly more balanced matchup, Alexander would have whaled all over Attila, and that the Hun would have tried his usual stuff, which Alexander was born to counteract and defeat, and that Attila wouldn't have known what hit him.