The primes, including the twin primes and the other prime pairs, are the building-blocks of the integers. Euclid’s proof of the infinitude of the primes has generally been regarded as elegant. It is a proof by contradiction, or, reductio ad absurdum, and it relies on an algorithm which will always bring in larger and larger primes, an infinite number of them. However, the proof is also subtle and has been misinterpreted by some with one well-known mathematician even remarking that the algorithm might not work for extremely large numbers. A long unsettled related problem, the twin primes conjecture, has also aroused the interest of many researchers. The author has been conducting research on the twin primes for a long time and had published a paper on them in an international mathematics journal in 2003. This informative paper presents some important facts on the twin primes which would be of interest to prime number researchers, with some remarks/reasons that point to the infinitude of the twin primes, including a reasoning which is somewhat similar to Euclid’s proof of the infinity of the primes; very importantly, 2 algorithms (refer to Appendix 3) for sieving out the twin primes from the infinite list of the integers are also presented, which would be of interest to cryptographers and even computer programmers.