the commercial and diplomatic influence of the Roman empire reached far beyond the borders of the empire. The Roman government also established various forms of over lordship over various states that were not part of any roman provinces. So the limits of Roman power are often hard to determine with certainty.
When Rome conquered Egypt in 30 BC, Egyptian trade with India became an important part of the Roman economy. Greek traders to India based on Egypt gradually become more and more Roman and all gained Roman citizenship in 211 AD.
An important stop on voyages to India, and a source of frankincense and myrrh, which were valuable products, was that "happy" place, Arabia Felix, or Yemen, South Arabia.
Emperor Augustus sent the Prefect of Egypt, Gaius Aeilius Gallus, on an unsuccessful expedition to Yemen in 26 BC.
The Romans also sometimes fought against various countries in the north of modern Sudan.
Emperor Nero sent an expedition to find the source of the Nile River, and it is uncertain how far south it got.
There were occasional wars with the Garamantes in the Sahara, and Emperor Septimius Severus captured their capital Garama.
Emperor Augustus also conquered a large part of modern Germany and established a province there, but the province was abandoned after a revolt.
In Britain, remains of Roman temporary marching camps and more or less permanent forts have been found far to the north in Scotland.
Tacitus says that a Roman fleet circled the north of Britain, proving it was an island, and reached the Orkney Islands, and perhaps even sighted Thule, wherever that was.
In the years 82 to 85, the Romans under Gnaeus Julius Agricola launched a campaign against the Caledonians in modern Scotland. In this context the Roman navy significantly escalated activities on the eastern Scottish coast. Simultaneously multiple expeditions and reconnaissance trips were launched. During these the Romans would capture the Orkney Islands (Orcades) for a short period of time and obtained information about the Shetland Islands. There is some speculation about a Roman landing in Ireland, based on Tacitus reports about Agricola contemplating the island's conquest, but no conclusive evidence to support this theory has been found.
You might find my answer here: https://www.quora.com/Which-Roman-emperors-set-foot-in-Britain9 interesting.
There is also the possibility of Roman military activity in Ireland.
In the east, the Roman Empire briefly annexed Iraq and Armenia, and there is an inscription by Roman Soldiers near Baku, Azerbaijan.
In the 1st century CE, the Romans organized two Caucasian campaigns and reached Baku. Near the city, in Gobustan, Roman inscriptions dating from 84–96 CE were discovered. This is one of the earliest written evidences for Baku.
It has even been claimed that Roman Legionaries might have reached Uzbekistan:
The first Roman embassy to China was in 166 AD:
The first group of people claiming to be an ambassadorial mission of Romans to China was recorded as having arrived in 166 AD by the Book of the Later Han. The embassy came to Emperor Huan of Han China from "Andun" (Chinese: 安敦; Emperor Antoninus Pius or Marcus Aurelius Antoninus), "king of Daqin" (Rome).
And there were later embassies from the Roman and eastern Roman or "Byzantine" empire.
The final recorded embassy arrived in 1091 AD, during the reign of Alexios I Komnenos (r. 1081–1118 AD); this event is only mentioned in passing.