During the reigned 1564-1616 what was the period called?

This is mostly the Tudor period - named after Henry VII, Henry Tudor. The name died out when Elizabeth I died childless. It is also in large part the Elizabethan period, since Elizabeth reigned from 1558 to 1603. The last 13 years are of the Stuart period, under King James I (VI of Scotland). Most of all, though, it is the Skaespearian period, since these dates define his lifetime.