When did Isaiah live?

The prophet Isaiah's time can be fixed by reference to the reigns of the Kings in which he prophesied.

The vision of Isaiah the son of Amoz, which he saw concerning Judah and Jerusalem in the days of Uzziah, Jotham, Ahaz, and Hezekiah, kings of Judah.

Thus the first period of his ministry was during the reigns of Uzziah 792-740 BC and Jotham 750-738 (as regent) and 738-732 as ruler. Isaiah's second period of work began in the reign of Ahaz from 735 - 715 and continued into the fifteenth year of Hezekiah's rule, after which he retired from public ministry.