The Artscroll Intro to Sanhedrin says the Rambam in Hil. Sanhedrin 4:11 implies that it is theoretically possible to renew the institution of semichah.
Based on this, R' Yaakov Beirav was ordained in 1538 and ordained 4 others, including R' Yosef Karo.
1. How is this renewed?
2. Given that it did renew in the 1500's, why did it stop?
Barry Epstein, Dallas, USA
There was a strong difference of opinion as to the Halachic validity of renewing the Sanhedrin. This was cause of significant controversy at the time.
Eventually, the dissenters prevailed, but not before the Beis Yosef (Rav Yosef Karo) was given Semichah by 200 of the greatest Sages of the time, which is what Rav Yakov Beirav, known as Mahari Beirav, required. This is the source for the acronym, "MaRaN," which stands for the words, "Masa'im Rabanan Nismach" (ordained by two hundred Sages).