The question as to when the Talmud was written down might seem like an elementary question that any schoolchild can answer – the Mishna was written down by Rabbi Yehudah HaNasi around the year 200 C.E., and about 300 years later around the year 500 C.E. the Talmud was redacted by Rav Ashi (352-427) and his student Ravina (d. 499).
The common perception, therefore, is that in 200 C.E. we had a collection of books called the Mishna and in 500 C.E. we had an even larger collection of writings called the Gemorra. Both works together formed what is known as the Talmud.