In this thesis we study the numerical range of quadratic operators and its generalizations. The main purpose of this thesis is to prove that the essential numerical range of a quadratic operator is a closed elliptical disc and present some remarkable results about c-numerical range of quadratic operator. Tso and Wu (1999) proved that the numerical range of a quadratic operator is an elliptical disc . Under some conditions this elliptical disc is open and is closed in some others. In 2007, Rodman and Spitkovsky prove the Tso-Wu theorem by different method. The benefit of their work was that they exactly used the same method for essential numerical range of quadratic operators and proved that the essential numerical range of a quadratic operator is a closed elliptical disc. They derived remarkable conclusions about c-numerical range of a quadratic operator. In this thesis first we work on Tso-Wu theorem and give its proof, then we also give the proof of the Rodman – Spitkovsky theorem. Then we prove the essential umerical range of quadratic operators and under specific conditions, c-numerical range of quadratic operators are elliptical disc.