Final answer:
The Mongol armies under Kublai Khan advanced into various regions such as the Kara-Khitan kingdom, the Khwarazmian Empire, and eastern Europe. However, they never invaded Europe fully.
Step-by-step explanation:
The Mongol armies under Kublai Khan advanced into multiple regions as part of their conquest. They moved westward into areas such as the Kara-Khitan kingdom, the Khwarazmian Empire, and northern China. They also continued into eastern Europe and the lands of the Holy Roman Empire, with scouts reaching the city walls of Vienna. However, despite their campaigns and victories, the Mongols never fully invaded Europe.