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:
, with 5129 digits, by Harvey Dubner.
, with 4932 digits, found by Indlekofer
and Ja'rai in 1994.
with 4030 digits, found by H. Dubner.
. Found by H. Dubner as well.
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.
References
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.