Abstarct: The interest in finding estimates on the Taylor-Maclaurin coefficients |a_2| and |a_3| for analytic and bi-univalent functions in the function class Sigma, caused that many mathematicians are active in this field. They couldn't find sharp estimates yet, but their aim are to improve these bounds. In this thesis, we introduce several subclasses of the function class Sigma of bi-univalent functions analytic in the open unit disc. Furthermore, we find estimates on the coefficients |a_2| and |a_3| for functions in these new subclasses.