This article attempts to critically review a rather extended field of ab initio calculations of mechanical properties of materials. After a brief description of the density functional theory and other approximations utilized in a majority of ab initio calculations, methods for predictions of elastic constants and moduli are presented. A relatively large space is devoted to computations of theoretical strength under various loading conditions. First we focus on results for perfect crystals and make an overview of advanced approaches to crystal stability. As case studies, elastic stability conditions defined according to both the adopted definition of elastic coefficients and the kind of applied loading are shown for isotropic tensile loading of molybdenum crystal and a model of microscopic deformation is illustrated for a soft phonon found in the dynamic stability analysis of isotropic loading of platinum crystal. Collected values of ideal strength under uniaxial/isotropic tension and simple shear for selected metallic and covalent crystals are discussed in terms of their comparison with available experimental data. Further attention is paid to results of studies on interfaces and grain boundaries. Applications of computed values of the moduli and the theoretical strength to prediction of intrinsic hardness and brittle/ductile behavior of crystalline materials and simulation of pop-in effect in nanoindentation tests are also included. Finally, remarks about possible topics for future ab initio studies and challenges for further development of computational methods are attached. (C) 2015 Elsevier Ltd. All rights reserved.