The appreciable progress has been achieved currently in the development of design principles, synthesis, investigations and predictions of various groups of ultra-incompressible and superhard materials. One of the recently proposed families of these promising materials is represented by the diborides of heavy 4d and 5d metals. Along with experiments, the theoretical ab initio methods, which involve no a priori assumptions about the electronic structure and intra-atomic interactions, are very effective approaches in the determination and prediction of structural, mechanical, magnetic, optical, dielectric and superconducting properties of such materials. This paper offers a review of the recent advances in theoretical understanding and predictions of the mechanical properties (including elastic (reversible) deformations, which are related to compressibility, as well as the effects of plastic (irreversible) deformations, which are related to hardness of materials) as obtained by means of ab initio calculations for a broad family of metal diborides MB2 (AlB2-like diborides and the recently discovered diborides of heavy 4d and 5d metals with "puckered" boron sheets) and their relations to electronic, cohesive and bonding characteristics of these materials. (C) 2011 Elsevier Ltd. All rights reserved.