The Twin Prime Conjecture is an interesting unsolved problem in Mathematics. Despite its apparent simplicity, there exist no conclusive answers to the question: "Are there infinitely many twin primes?" Some basic properties of prime numbers and apparent trends suggest that there are infinitely many primes. However, to this point, no one has found a way to prove this. In this book, we investigated the infinitude of the twin prime numbers using the most important prime sieving algorithms and we proved the correctness of the presented algorithms and computed the complexity of any sieving algorithm.