zwarrior:
Hello,
I have an Arduino Pro Mini 5V,16MHz board based on ATmega328. Can I run it safely and properly if I supply only 3.3V at the same 16MHz? Because, I want to use a battery to run my project and I have to use 3.3V supply. So, is it possible?
No, the safe operating voltage for 16 MHz is 4.5V ... 5.5V only.
Have a look into the datasheet of the Atmega328!
Safe operating voltage depends on the clock frequency you are using: The higher the clock frequency, the smaller is the range of the safe operating voltage. There is a diagram in the data sheet.
If you need to run at a lower voltage, this is only safe at a lower speed. You would have to replace the crystal oscillator against one with 8 MHz and the voltage regulator against one which regulates to 3.3V. Then you can go down to (I think) 2.7 V as a safe operating voltage. Look up the details in the datasheet!
In fact, also 16 MHz operation is possible at a lower voltage than 4.5V in some cases. Such like perhaps 3.8V, 3.7V or even 3.5V. But it is out of the specifications in the datasheet and it may only work at certain temperature ranges (perhaps room temperature) instead of the full temperature range of the controller. So it is not guaranteed by Atmel that the controller is working within its specifications if you don't operate within the voltage and clock frequency speed specifications.