Morphological shape transformations in biological systems often arise from patterned biochemical processes, which can produce mechanical forces either directly via molecular motors (e.g. the apical constriction in gastrulation) or indirectly via differential growth of connected tissues. The growth mismatch produces internal stresses, which can be released via shape transformations and mechanical instabilities. In this talk I will focus on mechanical instabilities that cause the wrinkling of bacterial biofilms and branching in developing lungs. Expanding bacterial biofilms experience compressive stress, because they stick to soft substrates and thus cannot expand freely. As biofilms grow the mechanical stress builds up and triggers the wrinkling instability, where the characteristic wavelength is determined from the biofilm thickness and from mechanical properties of the biofilm and substrate. The predicted wavelength agrees very well with the experimental measurements of Vibrio cholerae biofilm on the agar substrate. In the second part I will discuss our attempts to construct a minimal mathematical model that would capture the processes involved during the branching morphogenesis of lungs. Our starting model are simple tubes constructed from an individual epithelium on the inside of tube that is surrounded by the mesenchyme tissue and a thin layer of stiff smooth muscles. We investigate how patterned differential growth between the inner epithelium and outer mesenchyme tissue as well as the spatial pattern of smooth muscles lead to formation of new branches and their subsequent development.