We report colloidal synthesis of antimony (Sb) nanocrystals with mean size tunable in the 10-20 nm range and with narrow size distributions of 7-11%. In comparison to microcrystalline Sb, 10nm and 20nm Sb nanocrystals exhibit enhanced rate-capability and higher cycling stability as anode material in rechargeable Li-ion and Na-ion batteries. All three particle sizes of Sb possess high and similar Li-ion and Na-ion charge storage capacities of 580-640 mAh g-1 at moderate charging/discharging current densities of 0.5-1C (1C-rate is 660 mA g-1). At all C-rates (0.5-20C, e.g. current densities of 0.33-13.2 Ag-1), capacities of 20nm Sb particles are systematically better than for both 10nm and bulk Sb. At 20C-rates, retention of charge storage capacities by 10 and 20nm Sb nanocrystals can reach 78-85% of the low-rate value, indicating that rate capability of Sb nanostructures can be comparable to the best Li-ion intercalation anodes and is so far unprecedented for Na-ion storage.