Thin (D < 10 nm) nanowires are in principle promising for their application as catalysts and as elements of nanocomputers and quantum devices. To perform these tasks, their structure and properties must be stable at least at standard conditions. Using our technique based on the capture of small particles to the core of quantized vortices in superfluid helium, we synthesized nanowires made of various metals and alloys and investigated their thermal stability. The indium nanowires (D = 8 nm) were shown to be stable when heated to 100 degrees C, i.e., almost to the melting point, whereas the silver nanowires (D = 5 nm) disintegrated into traces of individual nanoclusters at 300 K. The gold and platinum nanowires also decomposed at temperatures more than twice as low as the melting point. A model is proposed to explain the premature decay of thin nanowires by unfreezing of the surface-atom mobility in combination with the anomalous dependence of the surface tension on the nanowire radius. Methods for improving the stability limits of thin nanowires by saturation of their surface with immobilized atoms as well as by surface oxidation have been proposed and experimentally tested.