There are a lot of places which romans have invaded, the 3 are italy, germany and great britian
In Europe: Italy, Portugal, Spain, France, Belgium, Holland south of the river Rhine, Southern Germany, Switzerland, Austria, part of western Slovakia, western Hungary, Slovenia, Croatia, Serbia, Bosnia & Herzegovina, Montenegro, Macedonia, Albania, Greece, Bulgaria, Romania, Moldova, a slither of Ukraine and England and Wales.
In Asia: Turkey, Cyprus, Armenia, Syria, Lebanon, Palestine, Jordan and the northern coast of the Red Sea in Saudi Arabia. At times it also included Iraq.
In Africa: Egypt, the coast of Libya, Tunisia, the coast of Algeria, and northern Morocco.
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The Romans conquered Greece, Romania, Egypt, Britain, France, Spain, Portugal, and the top part of Africa.
The Romans officially invaded Britain three times. Twice by Julius Caesar and once by Claudius.Read more: How_many_times_did_the_Romans_invade_Britain
the Romans built public baths
Yes, they did and waged a war against the Picts. However, they didn't stay long because to the Roman way of thinking, there was no benefit in holding the territory. There are Roman ruins in Southern and Mid-Scotland attesting to their presence.