Complex, three dimensional (3D) assemblies of nanomaterials provide sophisticated functions in even the most basic forms of life. Compelling opportunities exist for analogous 3D structures in man-made devices, but existing design options are highly constrained by comparatively primitive capabilities in fabrication and growth capabilities. Here we summarize a collection of advances that provides broad access to diverse classes of 3D architectures in advanced materials, including semiconductor nanomaterials, with characteristic dimensions that range from nanometers to centimeters and areas that span square centimeters or more. The approach relies on geometric transformation of two dimensional (2D) precursors into extended 3D layouts by controlled processes of compressive buckling. Experimental and theoretical studies demonstrate the capabilities in a diversity of structures with various forms and functions. The collective results establish unique possibilities for unusual classes of 3D microsystems technologies – as demonstrated in examples ranging from electronic microfliers, to sub-millimeter scale robots, and mesoscale biointerfaces.