During some years we have seen great progress in solving old problems in minimal surfaces theory. Among these problems are the Calabi-Yau s conjectures, dating from the 60s of last century. The first one stated that there were no complete minimal surfaces contained in a ball of R3, and the second one that all complete minimal surface should have an unbounded projection in each axes. In this work we pretend to review two examples that proof the falsity of the second conjecture. The first one was given by L. P. Jorge e F. Xavier (1980) and the second one by H. Rosenberg e E. Toubiana (1987). The first conjecture is also false. The first counterexample was given by N. Nadirashvili (1996) and it is also a counterexample to the conjecture of Hadamard, which stated that there were no complete bounded surfaces with negative Gaussian curvature. Development of Nadirashvilli s article is the main objective of this dissertation. The technique used in these three works is the use of the Enneper-Weierstrass Representation, combined with appropriate applications of Runge s theorem.