While we wait for experts to possibly weigh in, you might find this thread a good place to start. Axes are touched upon, among other weapons.

EDIT: The AutoMod has reminded me to name drop. /u/egregioustopiary's response is the one that caught my eye.

In Ancient Greece, while axes were known and used as tools, we have no evidence of them being used in battle. The typical close combat weapon of the Greeks was the spear - an ashen thrusting weapon about 6-8ft long with an iron leaf-shaped head and a bronze butt-spike. Greek heavy infantry used short swords as sidearms. The two main types were the leaf-shaped xiphos and the kopis ("chopper") or machaira, a recurve sabre meant for downward slashing.

While there are no examples of Greeks preferring axes as sidearms rather than swords, we know this was common among the Persians. Both infantry and cavalry could carry either a straight sword (called an akenakes), a machaira, or a one-handed axe (called a sagaris). You can see an ancient depiction of it here. The Greeks regarded this weapon as typical for the horsemen of the Sakai and the Skythians. In addition, they often depicted Amazons wielding battle axes of this type, which suggests that it was quintessentially exotic and barbarian to Greek eyes.

Mods, can we post links to videos of experts answering these questions?

Could you define ancient please? The short answer is that there is much less evidence of their use in the classical period compared to the early middle ages and later.