I can answer this for the West, but not the entire world.

In ancient Mesopotamia, we see male figures with beards but shaven heads. It's really in ancient Egypt that we see beardless men, while they still kept up fake beards for ritual purposes.

Stone blades that may be razors date back to 18,000 BC. Also used by various pre-metallic people are clam shells and shark's teeth.

The earliest razors we have are bronze, like a knife blade with a round or turned-back tip, very broad with a blunt back. The same people might use obsidian blades, too. The Hallstatt Culture of Europe (8th-6th C BC) has left bronze razors.

In the 6th C BC, Lucius Tarquinius Priscus introduced razors to Rome, according to Livy, but they didn't become common for another century.

The Romans were kind of strange in wanting to shave: most European/Mediterranean cultures keep the beard. However, since they became the norm for the Roman Catholic Church, the Church tended to declare fashions for beards heathen and all that, and try to get medieval men to wear clean chins and short hair like Romans.

SOURCE:

Boucher, 20,000 Years of Fashion (he's very good on ancient stuff, compared to most histories of fashion)