While @Fana's answer is correct for whether a US controller will work on an EU system, it might also be possible to repair your current controller rather than replacing it, (depending on what is wrong with the controller).

One of the more common problems (and the easiest to fix) is that the buttons don't work (or only work intermittently). The most common cause for this is degradation of the Rubber Domes that provide the button 'press' and 'depress'. After 20+ years, these domes can degrade to the point of splitting or flaking apart, meaning the button no longer 'springs' back into place, or doesn't actually make an electrical connection with the button at all.

Luckily, these rubber domes can be cheaply and easily replaced.

I've previously used Nintendo Repair Hut* to order replacement rubber domes. Despite the name they also do non-Nintendo consoles, here's the page for Mega Drive/Genesis repair supplies.

You can get replacement domes for both 3-button and 6-button Mega Drive/Genesis controllers, either set will run you about $2 USD, plus some shipping. The rubber domes look like this:

3-Button controller:

^{Click for larger image}

For the 3-button pad, The big dome with four buttons is for the D-Pad, and the other singular buttons are for the main 'A B C' controller buttons and Start button.

6-button controller:

^{Click for larger image}

The 6-button controller is a little different: the X Y Z, A B C buttons are now handled by the large rectangular piece. The D-Pad also only has one side-hole to orient it and hold it in place

_{* Note that I'm not affiliated with Nintendo Repair Hut, and that you can get these rubber domes from other suppliers as well, I'm just using them as the example as it's who I contacted to get my replacement rubber domes from.}

Replacement

To replace the domes, you'll need a Phillips-Head screwdriver to open up the Mega Drive/Genesis controller.

• Unplug and place the controller face-down
• Unscrew both screws on the back
• Carefully remove the back cover

• Lift off the circuit board & attached cable:

  
  ^{Click for larger image}

• Remove the old rubber domes
  • They may be slightly stuck, and they may rip/come apart in your hands. Be sure to remove all pieces.
• Place the new domes in the correct positions.
  • The D-Pad dome should be oriented so that the controller's plastic pins go through the holes on the D-Pad dome:
    
    ^{Click for larger image}
• Replace the circuit board and back of the controller, being sure to orient the cable to fit through the hole.
• Screw it back together
• Test the controller.

If the controller is now working better, congratulations!

References

• NintendoHut - for pictures of rubber domes
• Electrothoughts for internal pictures of the 3-button controller

Yes they will, all the genesis controllers have the same plug.

Fun fact, you can also use a Master System controller or a C64 controller which all have the same ports (although the Master system / C64 only have 2 trigger buttons).

You can use controllers from master system, Atari, and Mega Drive (including US genesis) of any contries on your Mega Drive.

The controlers are compatible between all old consoles (except of nintento). Evenn Atari controler (wich has less buttons than mega drive) will work for games like Sonic where you dont need all buttons.