What is a sieve? One possible answer is: a technique to extract information from a sequence A, for instance an interval of integers, to get properties of a smaller sequence, for instance the number of primes within this interval, or the number of prime twins. But what kind of information are we talking about? Well, usually information on how this sequence is distributed in some special arithmetic progressions. The fact is that these processes are usually very general, which implies that we can use them even in some tricky situations. As only to be expected, these processes also have limitations. With that being said, sieve methods have had a long history and it has been, practically, over a century since the birth of the so called Modern Sieve Theory. The Sieve Theory has had an extraordinary development and it has greatly evolved, to the point that it is now a powerful tool in Number Theory and in other branches of Mathematics, such as the Theory of Automorphic Forms (essentially after the great work of H. Iwaniec). Also, in applied fields of Number Theory, such as Cryptography and Algorithmic Number Theory, sieve methods are extremely important and there are many applications. We have direct applications, like finding all the prime numbers within certain boundaries or the construction of numbers free of large prime factors; as well as some indirect applications, like the fact that sieve methods and sieve techniques can yield valuable information about the distribution of “smooth numbers” in short intervals, and hence allows us to bound the running time of some factoring algorithms. To summarize, in this thesis some sieve methods are studied and some examples of their applications in Number Theory are also presented.