3.7V Rechargable battery as arduino power supply
For Li-Po or Li-Ion you'll need an appropriate charge circuit / battery controller.  These can be quite complex.  The battery must not go flat or it's dead for good.  The BQ2000T from Texas Instruments is a good (and popular) starting point for such a circuit.

The AVR on the Arduino can run happily at 3.7V.  If you have any level translation between the AVR and your board you would be able to ditch that as it would be no longer needed.

You'd normlly run at 3.3V not 3.7 or 3.8 as it's more of a standard voltage.  Are you sure that 3.8 is right for your board, and that's not the absolute maximum?