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New record prime (GIMPS): 2^{82,589,933}1 with 24,862,048 digits by P. Laroche, G. Woltman, A. Blosser, et al. (7 Dec 2018). 

The Largest Known Primes Database  The "Guinness book" of prime number records! Includes the 5000 largest known primes and smaller ones of selected forms (onepage summary) updated daily!  
Lists of Primes  The first 1,000 primes. The first 50,000,000 primes. Top 20 records (e.g., twin primes, Mersenne primes...) Lists of 300 digit primes. And much more!  
Finding primes, proving primality  Explains the mathematical theory behind how these record primes are found.  
How many are there?  Infinity, but How Big of an Infinity?  
The Largest Known Prime by Year: A Brief History  Discusses how big have the largest known primes been historically (and uses that to predict how big they will be)! 
 Conjectures and Open Problems
 A short list of conjectures and open problems relating to primes.
 The Riemann Hypothesis
 One of the most important conjectures in prime number theory. When (and if) it is proven, many of the bounds on prime estimates can be improved and primality proving can be simplified.
 Prime Curios!
 "Prime Curios!" is an exciting collection of curiosities, wonders and trivia related to prime numbers.
 Prime Glossary
 Definition of terms related to prime numbers and primality.
 Check a Number's Primality
 A simple routine to check most small numbers for primality.
 Important Discovery: "Primes in P"
 Primality can be tested in deterministic polynomial timethis is of great theoretical value; but of questionable practical value.
