The Moto X arrived at UK shores just yesterday and the phone's £380 launch price has already been reduced. Clove UK is offering a significant price cut to £300 and has the phone in stock, with today shipping.
Note that this is for a SIM-free 16GB version, the 32GB model will not be available in the UK.
Most other retailers still list the phone at the £380 price, sometimes even higher, and have not started shipping yet. Amazon UK does have the Moto X at £350, so it's worth keeping an eye if they'll drop the price further.
The price in the rest of Europe was initially announced as €430, but Amazon Germany has it at €350 (shipping on February 6). Both black and white models are available (in the UK the white one is exclusive to Phones4u) and so is the 32GB version (at €400).
In other European countries the Moto X 16GB is €400 (Spain) and even the full €420 (France, Italy). We'd wait a few days for the device to start shipping to see if there will be price cuts in those countries.