Materials and methods to achieve electronics intimately integrated on the surfaces of substrates with complex, curvilinear shapes are described. The approach exploits silicon membranes in circuit mesh structures that can be deformed in controlled ways using thin, elastomeric films. Experimental and theoretical studies of the micromechanics of such curvilinear electronics demonstrate the underlying concepts. Electrical measurements illustrate the high yields that can be obtained. The results represent significant experimental and theoretical advances over recently reported concepts for creating hemispherical photodetectors in electronic eye cameras and for using printable silicon nanoribbons/membranes in flexible electronics. The results might provide practical routes to the integration of high performance electronics with biological tissues and other systems of interest for new applications.