It seems that chariots were used across most of the ancient world for about 2,000 years, before falling out of favor. Wikipedia states "Chariots had lost their military importance by the 1st century AD," but the article doesn't really say why (unless I missed it).
Was there some technological or tactical innovation that made them obsolete? Maybe an economic reason?
It's a rather surprising explanation. The technological development was from chariotry to horseback riding.
Strange as it may seem, humans in the Near East developed chariots for purposes in warfare around 2000 BCE, before the use of mounted soldiers. The idea of horseback riding had been around for some time, and chariots would have been more expensive, yet mounted troops don't appear until the Scynthians in the 7th century. The common explanation is that early horses were much smaller and it took a couple of millennia to breed any strong enough to carry men in battle.
By the first century CE, the economic (cheaper) and tactical advantages of cavalry (they are more maneuverable and able to be used on varied terrain) made chariots obsolete.
Sources to check out: DW Anthony - The Horse, the Wheel, and Language
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