The mean-field approximation (MFA) is the basis of what we use to analyze the molecular properties created by the complex correlations among the electrons. The aim of this chapter is to provide the details of the conventional methods for constructing the mean field in the wave-function theory and in the density-functional theory (DFT). Section 1.1 describes an overview of the MFAs in these theories, which identifies the problems of the approximations in each of the theories. A novel approach developed to solve the problem in the DFT, which constitutes the major issue of this book, is also reviewed in this section. In Sect. 1.2, the MFA termed Hartree-Fock (HF) method of the wave-function theory is formulated in detail for later discussion. In the framework of the Kohn-Sham DFT (KS-DFT), the mean field is directly given by the functional derivative of the exchange-correlation functional \(E_{xc}[n]\) with respect to the electron density n. The theoretical foundations underlying the KS-DFT are provided in Sect. 1.3. The guiding principles for the developments of the approximate functionals used in the KS-DFT are also shown with emphasis on their physical meanings. An approach referred to as optimized-effective potential (OEP) offers a method to obtain the KS effective potential which minimizes an implicit energy functional of the density n. The method to implement the HF-OEP will be discussed in the last section. The inverse Kohn-Sham (inv-KS) method is another approach to construct the mean field from a given electron density. The relationship between the HF-OEP and the inv-KS for the HF density will also be discussed. In this chapter, the atomic units (a.u.) are used unless otherwise stated.