Surfaces are at the cutting edge of materials science. Molecules of environmental gas or liquid come in contact with a material at its surface. To understand how materials function, and how to produce and enhance them, we must understand surfaces. This article focuses on the methodologies used to calculate the electronic structure of crystalline solids and their surfaces from a theoretical physicist‘s point of view. Theorists are challenged to predict the atomic structure of a material from first principles, i.e. by using only the atomic number of each chemical element (its position in the periodic table of elements) and the fundamental laws of quantum mechanics as input. The intricacies of this task are illustrated by a discussion of the crystal structure of carbon and silicon, two elements from the same column of the period table of chemical elements that prefer to crystallize in the graphite and in the diamond structure respectively. Density functional theory has become the main workhorse for dealing with complex surface structure. Dating back to seminal achievements of Walter Kohn and collaborators in the mid sixties of the last century, its great popularity is based on Kohn‘s famous theorem; stating that `the electronic density determines everything‘. This implies that all intricacies of a quantum-mechanical many-particle system can be reproduced from the density alone, a quantity that is moderately sized even for systems of many atoms, and can be handled with affordable effort on present-day computers. This approach is exact in principle, but becomes tractable only by (barely controllable) approximations concerning the quantum mechanical many-body interactions between the electrons. This is contrasted with alternative methodologies; in particular, the Quantum Monte Carlo method, for calculating directly the many-particle electron wave function, an object whose information complexity far exceeds that of the electron density. While this method uses an analogy to statistical physics and thus produces results only within a statistical error bar, it allows for systematic improvements of these results to (almost) any desired precision. The capabilities of both the density functional and the Quantum Monte Carlo method are exemplified by calculations of the surface structure of carbon and silicon surfaces, and by recent calculations of hydrogen adsorption on the silicon surface. While density functional theory reproduces the experimentally observed `buckling‘ of the silicon surface, the Quantum Monte Carlo method additionally yields chemical binding energies and energy barriers in better agreement with experiment. Based on this knowledge, an improved understanding of, for example, the mechanisms of chemical reactions at surfaces can be achieved.