Can my UNO or MEGA run of of 15v dc safely???

In doing a project which uses an UNO to drive a laser galvo.
The galvo has a 15v power supply. It would be great to tap into that to power the arduino rather than getting a second power supply since space is limited in the project box.

Is it safe to run the arduino off of the same 15v power supply??? Ive never used an arduino with anything other than 5V...

Summary
Microcontroller ATmega328
Operating Voltage 5V
Input Voltage (recommended) 7-12V
Input Voltage (limits) 6-20V

At over 12V the 5V regulator will likely get very warm even without pulling much load.

You can look up the MEGA here, the practice will do you good. Without it you are dependent on others.

Here is another good place to find answers to what 90% of questions seem to be.

s it safe to run the arduino off of the same 15v power supply

I would say yes. The major limiting factor is the voltage rating of the electrolytic capacitors on the input. On my Uno these are 25V so 15V should pose no problems.

Yes the regulator will run hotter but this is unlikely to be a problem unless you draw a lot of current off the 5V lines. For a discussion of how much hotter see:-
http://www.thebox.myzen.co.uk/Tutorial/Power.html
and the specific issue of the arduino board is shown here:-
http://www.thebox.myzen.co.uk/Tutorial/Power_Examples.html

Could always add a 9V regulator for the arduino, couldn't you? Would be easy insurance to me.

Drop the voltage using a series of diodes (properly rated), i'm also about to use them to reduce a 5v supply to 4.5v range.

I have been running my arduino from 15V all day today developing a project that needs 15V to drive some coils, there has been no noticeable heating on the regulator.