Spectrometers are widely used tools in chemical and biological sensing, material analysis, and light source characterization. Traditional spectrometers use a grating to disperse light, and the spectral resolution scales with the optical pathlength, imposing a trade-off between device size and resolution. To develop a compact spectrometer without sacrificing resolution we turn to a multimode fiber, where a long pathlength is easily achieved in a small footprint by coiling the fiber. Of course, replacing the grating with a multimode fiber also requires the spectrometer to operate according to a different paradigm. Here, we use the wavelength-dependent speckle patterns formed by interference between the guided modes of a multimode fiber as a fingerprint to identify the input spectra. The spectral resolution is then determined by the change in wavelength required to produce an uncorrelated speckle pattern, which scales inversely with the length of the fiber. Using a 100 meter long fiber, we were able to resolve two lines separated by only 1 pm near λ=1500 nm. We also achieved broad-band operation with a 4 cm long fiber, covering 400 nm – 750 nm with 1 nm resolution. Since the fiber can be coiled into a small volume, the entire device remains compact, lightweight, and low-cost. We applied the same approach of using wavelengthdependent speckle patterns to build an on-chip spectrometer where the device size is particularly limited. In this case, we introduced multiple scattering in a disordered structure by etching holes in a silicon membrane to increase the effective optical pathlength in a fixed footprint. We achieved 0.75 nm resolution at λ=1500 nm in a 25 μm radius spectrometer. Such a high-resolution on-chip spectrometer could enable compact, low-cost spectroscopy for portable sensing or increasing lab-ona-chip functionality.