It could be a few things that would cause this, but the first thing I would check is it being an app you recently installed, or recently updated. Try and see if the phone will start, and stay on, in "Safe Mode". If it will, then you know for sure that an app or other file on the device is causing you problem.
To get into "Safe Mode" please do the following.
1. First turn off the phone.
2. Next press and hold "Power" until you see the manufacturers logo on the screen, then release the power button.
3. "Quickly" press "Volume Down" and hold it for few seconds until you see the message "Safe mode: ON" or something similar.
4. If you wanted disable safe mode, simply repeat the steps above, and you should see "Safe Mode: Off" rather than On.
If it still turns off, even in "Safe Mode", then there is something else going on. Perhaps something loose inside the device. In such a case I would be careful about messing with the phone yourself, at least if it is still under warranty, as that would void the warranty. In that case it would be best to contact your carrier, or the manufacturer and see if it can be repaired or replaced.