Abstract : Deciding if a system of polynomial equations has a solution over Q is one of the oldest questions in number theory and over the years it has proved to be a real challenge. One of the first     approaches to this question is to prove existence of solutions over bigger and  »simpler » fields (i.e. Q_p, R) and then to try to  »restrict » the solutions, this is known as the local-global principle. In         this talk, I will present the local-global principle, some counter examples to this principle, and further approaches to finding rational solutions.