Flexible, thin-film electronic and optoelectronic devices typically involve a trade-off between performance and fabrication cost(1-3). For example, solution-based deposition allows semiconductors to be patterned onto large-area substrates to make solar cells and displays, but the electron mobility in solution-deposited semiconductor layers is much lower than in semiconductors grown at high temperatures from the gas phase(4). Here, we report band-like electron transport in arrays of colloidal cadmium selenide nanocrystals capped with the molecular metal chalcogenide complex(5,6) In(2)Se(4)(2-), and measure electron mobilities as high as 16 cm(2) V(-1) s(-1), which is about an order of magnitude higher than in the best solution-processed organic(7) and nanocrystal(8) devices so far. We also use CdSe/CdS core-shell nanoparticles with In(2)Se(4)(2-) ligands to build photodetectors with normalized detectivity D* > 1 x 10(13) Jones (I Jones = 1 cm Hz(1/2) W(-1)), which is a record for II-VI nanocrystals. Our approach does not require high processing temperatures, and can be extended to different nanocrystals and inorganic surface ligands.