The ancient Romans and Greeks did not use stirrups that first appeared in China. The oldest image is from 322 CE.
Stirrups are considered to be one of the greatest military discoveries in history, prior to the invention of gunpowder. It is often believed that they were among the most important military inventions after chariots and saddles.
With the help of stirrups, the rider could be more stable in the saddle and therefore fight the infantry better and more confidently; he was more manoeuvrable on the battlefield and during the campaign.
The Romans used saddles that had a special construction. They had four corners surrounding the seated person. This way the rider had a reasonably stable position despite the lack of stirrups. A great example is the coin of Quintus Labienus from around 39 BCE, on the reverse of which you can see a saddled horse.
