Can low battery kill arduino or leds?

I want to make a simple led project. Arduino will be powered by battery (leds will have external power supply from the same battery).
My question is what can happen if battery discharges? I don't have a lot of knowledge in electronics but I often read to be careful with too low voltage. So, can I damage/burn my arduino or leds if battery gets low? Or maybe it will just safely stop working?

The chip has low-voltage detection and simply shuts down if the voltage drops too low - however for the Uno I think the low voltage threshold is set to 2.7V, which is lower than a 16MHz Uno will reliably work at.

However with these low voltage devices nothing bad will happen as there's very little power availble to do damage. Especially as things like LEDs with series resistors can only use less power at lower voltage.

You can program the low voltage threshold to various values to suit your application (in the datasheet the phrase "brown-out detection" or "BOD" is used.

Context?
With rechargeable batteries especially lithium, discharging to a too low voltage will damage the batteries. Is that what you read?

Brown-outs were a problem for old TV and such. P=I*E if the power draw remains at say 500 watts and the voltage drops the current must increase. Many older units would burn up as in an instant current shot towards infinity and the system shut down. I think most modern units are not as senative as older units. I guess it is mathematically possible but very likely.