A general-purpose all-fiber spectrometer is demonstrated to overcome the trade-off between spectral resolution and bandwidth. By integrating a wavelength division multiplexer with five multimode optical fibers, we have achieved 100 nm bandwidth with 0.03 nm resolution at wavelength 1500 nm. An efficient algorithm is developed to reconstruct the spectrum from the speckle pattern produced by interference of guided modes in the multimode fibers. Such an algorithm enables a rapid, accurate reconstruction of both sparse and dense spectra in the presence of noise.