Nalanda was the ancient Indian university that flourished during the Gupta period. More precisely, Nalanda Mahavihara was a large Buddhist monastery and residential center of learning in Magadha, near modern Rajgir in Bihar.
The institution is traditionally associated with Gupta ruler Kumaragupta I, who is said to have established it around 427 CE. Nalanda expanded under Gupta patronage during the fifth and sixth centuries, although its supporters included rulers and donors from several religious backgrounds. Students and scholars studied Buddhist philosophy, logic, grammar, medicine, mathematics, and other branches of knowledge.
Nalanda attracted visitors from across Asia, including monks and students from China, Tibet, Korea, and Central Asia. It continued to prosper under later rulers, especially Harsha and the Pala dynasty, and operated for centuries. Calling it a “university” is useful but somewhat modern, because Nalanda was fundamentally a monastic educational complex rather than a university organized exactly like a modern institution. Vallabhi and Vikramashila were also important centers of learning, but Nalanda is the canonical answer for this question.