Galileo Positioning System signal receiving I want to determine my arduino's position. However, I want to use the European Galileo system instead of the US-American GPS. 

Therefore, I searched for any possibilities receiving the Galileo signals. I figured out, that u-blox has some receivers: NEO 8 or the LEA 6 (firmware update required). However I did not find any arduino breakout board (, etc.) that uses these u-blox sensors or other Galileo-compatible hardware.

Does anyone know, how I could use the Galileo Position System?
You need the hardware and a library. Most GPS modules use a serial communication, and I think that also the Galileo compatible boards use serial communication. That makes them compatible with most microcontrollers and also with Arduino boards.

I could find a few shields that can be attached to an Arduino board.
This one is in development at the moment: Kickstarter shield with Galileo
This one is compatible via the serial pins: GA-001 Multi GNSS Ublox
There is a "NEO-M8N" shield and maybe one or two others.

All those boards are very new, and I don't know if they come with a library for an Arduino. Perhaps you have to wait a few months and then it should be a lot easier to buy a complete functioning module with a library.

Before buying a module, confirm that there is a library for Arduino or good documentation to write your own library.