The nation that was invaded by the Soviet Union in the late 1970's was Afghanistan.
Explanation :
The Soviet Union has the southern borders with Afghanistan. They are concerned about the rise of Islamic extremism and Western encroachment on southern borders. The then-president Hafizullah Amin of Afghanistan was negotiating with the USA for settling some internal issues.
This caused a concern for the Soviet Union and in the year 1979 they invaded Afghanistan to have control over the government and they established their leader President Babrak Karmal. This invasion was considered to be the final one because the Soviet Union was started to disintegrate in the year 1991.