Cavalry
The most notable of these was the Battle of Cannae, where he inflicted a catastrophic defeat on the Romans. At about the same time the Romans began to recruit foreign auxiliary cavalry from among Gauls, Iberians, and Numidians, the last being highly valued as mounted skirmishers and scouts (see Numidian cavalry). Julius Caesar had a high opinion of his escort of Germanic mixed cavalry, giving rise to the Cohortes Equitatae. Early emperors maintained an ala of Batavian cavalry as their personal bodyguards until the unit was dismissed by Galba after the Batavian Rebellion.
For the most part, Roman cavalry during the early Republic functioned as an adjunct to the legionary infantry and formed only one-fifth of the standing force comprising a consular army. Except in times of major mobilisation about 1,800 horsemen were maintained, with three hundred attached to each legion. 
The relatively low ratio of horsemen to infantry does not mean that the utility of cavalry should be underestimated, as its strategic role in scouting, skirmishing, and outpost duties was crucial to the Romans' capability to conduct operations over long distances in hostile or unfamiliar territory. On some occasions Roman cavalry also proved its ability to strike a decisive tactical blow against a weakened or unprepared enemy, such as the final charge at the Battle of Aquilonia.
After defeats such as the Battle of Carrhae, the Romans learned the importance of large cavalry formations from the Parthians. 
At the same time heavy spears and shields modelled on those favoured by the horsemen of the Greek city-states were adopted to replace the lighter weaponry of early Rome. These improvements in tactics and equipment reflected those of a thousand years earlier when the first Iranians to reach the Iranian Plateau forced the Assyrians to undertake similar reform. Nonetheless, the Romans would continue to rely mainly on their heavy infantry supported by auxiliary cavalry.

Late Roman Empire and the Migration Period: In the army of the late Roman Empire, cavalry played an increasingly important role. The Spatha, the classical sword throughout most of the 1st millennium was adopted as the standard model for the Empire's cavalry forces. By the 6th century these had evolved into lengthy straight weapons influenced by Persian and other eastern patterns. Other specialist weapons during this period included javlins, long reaching lancers, axes and maces.
The most widespread employment of heavy cavalry at this time was found in the forces of the Iranian empires, the Parthians and their Persian Sasanian successors. Both, but especially the former, were famed for the cataphract (fully armored cavalry armed with lances) even though the majority of their forces consisted of lighter horse archers. The West first encountered this eastern heavy cavalry during the Hellenistic period with further intensive contacts during the eight centuries of the Roman–Persian Wars. At first the Parthians' mobility greatly confounded the Romans, whose armoured close-order infantry proved unable to match the speed of the Parthians. However, later the Romans would successfully adapt such heavy armor and cavalry tactics by creating their own units of cataphracts and clibanarii.
The decline of the Roman infrastructure made it more difficult to field large infantry forces, and during the 4th and 5th centuries cavalry began to take a more dominant role on the European battlefield, also in part made possible by the appearance of new, larger breeds of horses. The replacement of the Roman saddle by variants on the Scythian model, with pommel and cantle, was also a significant factor as was the adoption of stirrups and the concomitant increase in stability of the rider's seat. Armored cataphracts began to be deployed in Eastern Europe and the Near East, following the precedents established by Persian forces, as the main striking force of the armies in contrast to the earlier roles of cavalry as scouts, raiders, and outflankers.
The late-Roman cavalry tradition of organized units in a standing army differed fundamentally from the nobility of the Germanic invaders—individual warriors who could afford to provide their own horses and equipment. While there was no direct linkage with these predecessors the early medieval knight also developed as a member of a social and martial elite, able to meet the considerable expenses required by his role from grants of land and other incomes.

Asia: 

Central Asia: Xiongnu, Tujue, Avars, Kipchaks, Khitans, Mongols, Don Cossacks and the various Turkic peoples are also examples of the horse-mounted groups that managed to gain substantial successes in military conflicts with settled agrarian and urban societies, due to their strategic and tactical mobility. As European states began to assume the character of bureaucratic nation-states supporting professional standing armies, recruitment of these mounted warriors was undertaken in order to fill the strategic roles of scouts and raiders.
The best known instance of the continued employment of mounted tribal auxiliaries were the Cossack cavalry regiments of the Russian Empire. In Eastern Europe, and out onto the steppes, cavalry remained important much longer and dominated the scene of warfare until the early 17th century and even beyond, as the strategic mobility of cavalry was crucial for the semi-nomadic pastoralist lives that many steppe cultures led. Tibetans also had a tradition of cavalry warfare, in several military engagements with the Chinese Tang dynasty (618–907 AD).

Khanates of Central Asia: 

East Asia: 

China: Further east, the military history of China, specifically northern China, held a long tradition of intense military exchange between Han Chinese infantry forces of the settled dynastic empires and the mounted nomads or "barbarians" of the north. The naval history of China was centered more to the south, where mountains, rivers, and large lakes necessitated the employment of a large and well-kept navy.
In 307 BC, King Wuling of Zhao, the ruler of the former state of Jin, ordered his commanders and troops to adopt the trousers of the nomads as well as practice the nomads' form of mounted archery to hone their new cavalry skills.

The adoption of massed cavalry in China also broke the tradition of the chariot-riding Chinese aristocracy in battle, which had been in use since the ancient Shang dynasty (c. 1600–1050 BC). By this time large Chinese infantry-based armies of 100,000 to 200,000 troops were now buttressed with several hundred thousand mounted cavalry in support or as an effective striking force. The handheld pistol-and-trigger crossbow was invented in China in the fourth century BC; it was written by the Song dynasty scholars Zeng Gongliang, Ding Du, and Yang Weide in their book Wujing Zongyao (1044 AD) that massed missile fire by crossbowmen was the most effective defense against enemy cavalry charges.

On many occasions the Chinese studied nomadic cavalry tactics and applied the lessons in creating their own potent cavalry forces, while in others they simply recruited the tribal horsemen wholesale into their armies; and in yet other cases nomadic empires proved eager to enlist Chinese infantry and engineering, as in the case of the Mongol Empire and its sinicized part, the Yuan dynasty (1279–1368). The Chinese recognized early on during the Han dynasty (202 BC – 220 AD) that they were at a disadvantage in lacking the number of horses the northern nomadic peoples mustered in their armies. Emperor Wu of Han (r 141–87 BC) went to war with the Dayuan for this reason, since the Dayuan were hoarding a massive amount of tall, strong, Central Asian bred horses in the Hellenized–Greek region of Fergana (established slightly earlier by Alexander the Great). Although experiencing some defeats early on in the campaign, Emperor Wu's war from 104 BC to 102 BC succeeded in gathering the prized tribute of horses from Fergana.
Cavalry tactics in China were enhanced by the invention of the saddle-attached stirrup by at least the 4th century, as the oldest reliable depiction of a rider with paired stirrups was found in a Jin dynasty tomb of the year 322 AD. The Chinese invention of the horse collar by the 5th century was also a great improvement from the breast harness, allowing the horse to haul greater weight without heavy burden on its skeletal structure.

Korea: The horse warfare of Korea was first started during the ancient Korean kingdom Gojoseon. Since at least the 3rd century BC, there was influence of northern nomadic peoples and Yemaek peoples on Korean warfare. By roughly the first century BC, the ancient kingdom of Buyeo also had mounted warriors. The cavalry of Goguryeo, one of the Three Kingdoms of Korea, were called Gaemamusa (개마무사, 鎧馬武士), and were renowned as a fearsome heavy cavalry force. King Gwanggaeto the Great often led expeditions into the Baekje, Gaya confederacy, Buyeo, Later Yan and against Japanese invaders with his cavalry.

In the 12th century, Jurchen tribes began to violate the Goryeo–Jurchen borders, and eventually invaded Goryeo Korea. After experiencing invasion by the Jurchen, Korean general Yun Kwan realized that Goryeo lacked efficient cavalry units. He reorganized the Goryeo military into a professional army that would contain decent and well-trained cavalry units. In 1107, the Jurchen were ultimately defeated, and surrendered to Yun Kwan.