There has been much debate over this frankly odd question over the years.

The most recent estimation in 1989 places the geographical centre of Europe at 54°54′N 25°19′E.

This is 26 km north of the Lithuanian capital Vilnius. A white granite monument was errected at this location in 2004.

Whimsically, the Open Air Museum of the Centre of Europe is nearby which contains the world's largest sculpture made of TV sets, now partially collapsed.

I've heard about other locations, like Krahule in Slovakia, Cihost in the Czech Republic, Rakhiv (or Dilove) in Ukraine and Bernotai in Lithuania.
The differences in opinion exist because of different ways of calculating where the point is. There are also disagreements about what the extreme points of Europe are (whether to consider the mainland ones or not).