In this colloquium we present how geometrical shape affects the mechanical properties of thin solid shells and how buckling instabilities change the geometry of periodic microstructures in materials. Using methods rooted in statistical mechanics, we find that random shape fluctuations and thermal excitations of thin solid sheets significantly modify their mechanical properties. Sheets subject to such fluctuations are much harder to bend, but easier to stretch, compress and shear. In spherical shells thermal fluctuations also effectively produce negative surface tension, which is equivalent to applying external pressure. We find that small spherical shells are stable, but for sufficiently large shells this thermally generated “pressure” becomes big enough to crush spherical shells. Finally, we show how methods from solid-state physics can help us deduce the geometry of buckled periodic microstructures. Buckling instabilities can change the microstructure symmetries, including a spontaneous chiral symmetry breaking, which drastically modifies the material properties.