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Even if the leading role of mechanical forces in tissue growth and remodeling is well admitted, it is only recently that the effects of physical forces on gene expression and cell dynamics have been more extensively investigated. In this multiscale context, developing in silico models of cells and tissues is necessary to understand how biochemical and biomechanical cell signaling pathways interplay to specify a wide range of cellular dynamical processes and tissue architectures. In this paper, we review modeling approaches exemplifying how mechanical instabilities may drive emergent biological processes at different scales. At the cell level,we discuss the self-organized character of cell protrusions dynamics and the associated oscillatory cell shape changes modulated by the cell cortex contractility. At the cell population level, we analyze how the in vitro morphogenesis of endothelial cell networks can be enhanced or inhibited by modifications of the mechanical homeostasis of the cell/extracellular matrix medium. We finally outline how our understanding of such mechanical instabilities may support tissue engineering approaches in which transduction of mechanical forces has to be optimized in order to obtain specific cellular responses and tissue functional properties.