The two largest known twin primes are . with 11713 digits, found by Indlekofer and Ja'rai in November, 1995. They are also the first known gigantic twin primes (primes with at least 10,000 digits).
Recent record holders are:
The two largest Sophie Germain primes (i.e. p and 2p+1 are both primes) are and q=2p + 1, found by Harvey Dubner, in October 3, 1995.
B. K. Parady and J. F. Smith and S. E. Zarantonello, Smith, Noll and Brown. Largest known twin primes. Mathematics of Computation, vol.55, 1990, pp. 381-382.