This thesis is about pricing European options using a Fourier-based numerical method called the COS method under the rough Heston model. Besides examining the efficiency and accuracy of the COS method for pricing options under the rough Heston model, it is also investigated if the rough Heston model produces the advantages of the so-called rough volatility models. To do so, the characteristic function of the rough Heston model is derived, and the COS method for the rough Heston model and also a Monte Carlo simulation scheme is introduced. Throughout the thesis, the theoretical background of the rough Heston model, the numerical techniques and some numerical experiments on European option prices and implied volatility behaviors are presented. Also, a calibration of the rough Heston model is performed using Artificial Neural Networks. As a result of this thesis, pricing of European options using COS method is succeeded. Moreover, it is shown that the rough Heston model produces the rough volatility behaviors as expected.