The design of an instrument employing a capacitive transducer for non-contacting measurements of displacement and vibration is considered, with the aim of combining high resolution and linear response over a wide range. The three-terminal transducer in which movement of the earthed workpiece alters the capacitance between two non-earthed electrodes, is preferred to the two-terminal transducer, in which a capacitance to earth is measured. A two-dimensional theoretical analysis leads to practical design formulae and methods of construction are discussed in detail. A typical transducer 12 mm in diameter can give a sensitivity constant to within 3% and a maximum error of 3 mu m over a range of 500 mu m, or 1% and 1 mu m over 250 mu m; the resolution is determined either by electrical noise at 1 nm for a bandwidth of 1 Hz, or by ambient changes at <50 nm for 1 degrees C change in temperature; vibrations of frequency <10kHz can be observed; performance with typical non-planar workpieces is quoted, and the possibility of measuring displacements of dielectrics is indicated. Transducers with a logarithmic response are described.