Recently some efficient numerical methods based on Fourier Analysis were proposed. In this thesis, I summarize the Carr & Madan method and Fang & Oosterlee method respectively. Furthermore, I proposed new methods and numerical schemes to improve and develop their methods. Carr & Madan method is the first method to use Fast Fourier Transform algorithm on option pricing, and the overall complexity is O(N log N). Although this method is fast, it has a quite large error when the strike prices are small. My main work in this thesis is to develop a new method which combines the quadrature technique and their method. The method is accurate and stable even the strike prices are small. The theoretical analysis shows that the overall complexity of this new method is still O(N log N). The main idea of Fang & Oosterlee method is to price the option based on Fourier cosine expansion, the convergence rate of this method is exponential and the computational complexity is linear. The disadvantage of their method is being inefficient when the option prices are calculated with many different strike prices. My main work is to develop a new efficient method which is based on their method. Moreover, I apply the Fourier sine expansion and Fourier series expansion to the option pricing, then I compare those methods in this thesis. In Chapter 1, Section 1.1, I introduce some basic knowledge of European options. Section 1.2, I give the definition of L´evy model, and show some Exponential L´evy models for the financial markets, I also give the characteristic function of these models. Section 1.3, I change the pricing problems to quadrature problem, and the definition of characteristic function are also obtained. In Chapter 2, Section 2.1, I show the basic idea of Fourier series, cosine and sine expansions, the definition of algebraic convergence rate, geometry convergence rate and some convergence theorems. Section 2.2, I will give the definitions of Fourier Transform and Inverse Fourier Transform. Section 2.3, I introduce the idea of Fast Fourier Transform (FFT), and some useful theorems of FFT. In Chapter 3, Section 3.1, I summarize the Carr & Madan method and point out the problem of this method. Section 3.2, I give the error analysis of Carr & Madan method which comes from their paper i.e. [4]. Section 3.3, I propose our method which combines the quadrature technique and Carr & Madan method, and our method is more accurate and stable. Section 3.4, the Numerical Experiments show the comparison of our method and Carr & Madan method in some models which are given in Section 2.2. In Chapter 4, Section 4.1, I summarize the Fang & Oosterlee method. Section 4.2, I will give the error analysis of Fang & Oosterlee method which comes from their paper i.e. [9]. Section 4.3, I propose our method which increases the efficiency of their method for the option prices calculated with many different strike prices. I also applied other expansions which are given in Section 2.1 to European option pricing. Section 4.4 is the Numerical Experiments.