Optical spectroscopy is a fundamental tool in numerous areas of science and technology. Much effort has focused on miniaturizing spectrometers, but thus far at the cost of spectral resolution and broad operating range. Here we describe a compact spectrometer that achieves both high spectral resolution and broad bandwidth. The device relies on imaging multimode interference from leaky modes along a multimode tapered optical fibre, resulting in spectrally distinguishable spatial patterns over a wide range of wavelengths from 500 to 1,600 nm. This tapered fibre multimode interference spectrometer achieves a spectral resolution down to 40 pm in the visible spectrum and 10 pm in the near-infrared spectrum (corresponding to resolving powers of 10(4)-10(5)). Multimode interference spectroscopy is suitable in a variety of device geometries, including planar waveguides in a broad range of transparent materials.