Fast radio bursts (FRBs) at cosmological distances have recently been discovered, whose duration is about milliseconds. We argue that the observed short duration is difficult to explain by giant flares of soft gamma-ray repeaters, though their event rate and energetics are consistent with FRBs. Here we discuss binary neutron star (NS-NS) mergers as a possible origin of FRBs. The FRB rate is within the plausible range of NS-NS merger rate and its cosmological evolution, while a large fraction of NS-NS mergers must produce observable FRBs. A likely radiation mechanism is coherent radio emission like radio pulsars, by magnetic braking when magnetic fields of neutron stars are synchronized to binary rotation at the time of coalescence. Magnetic fields of the standard strength (~ 10^{12-13} G) can explain the observed FRB fluxes, if the conversion efficiency from magnetic braking energy loss to radio emission is similar to that of isolated radio pulsars. Corresponding gamma-ray emission is difficult to detect by current or past gamma-ray burst satellites. Since FRBs tell us the exact time of mergers, a correlated search would significantly improve the effective sensitivity of gravitational wave detectors.
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