This lecture by Prof. Nicola Marzari (EPFL/PSI) serves as the first module of the 2025 PSI course "Electronic-structure simulations for user communities at large-scale facilities." Explore the fundamentals of electronic-structure theories, with a focus on Density Functional Theory (DFT) and advanced methods beyond it. The comprehensive presentation covers essential theoretical frameworks that form the foundation for computational materials science and quantum chemistry simulations. Gain insights into the mathematical and physical principles that enable accurate modeling of electronic properties in materials. This module provides crucial knowledge for researchers and users of large-scale facilities interested in applying electronic-structure methods to their scientific investigations. For more information about the complete course, visit the PSI course details page at https://indico.psi.ch/event/17436/.