Is it possible to power STM32 Nucleo Boards from a 3.7 Lithium Ion battery?
STM32 Nucleo boards use an LD39030 to drop the 5V power input to 3.3V. This regulator has a typical dropout voltage of 0.2V at 500mA (proportionally less at lower current), so the Li-ion battery would have to stay above 3.5V at maximum board current. 

Provided you use a suitably sized battery it should work fine, though the effective battery capacity may be reduced. Most li-ion cells can produce over 3.5V for ~70% of their actual capacity at low current draw.

Example: Sanyo/Panasonic NCR18650GA 3500mAh 



For battery health you should avoid using more than 80% anyway, so 70% is not bad.