Are all Europeans the same race?

While I am not arguing that Europeans are white, yes they are all white, but they vary in appearance. I met this Greek girl a little while ago and she was quite dark, I thought she was From the middle east or even India. I also have met people from Spain who have a peculiar look that is more ethnic, rather than like white people of North America (Pedro Almodovar, Penelope Cruz, Rafael Nadal, Julio Iglesias, Almudena Fernandez) wouldn't really turn heads in South America, same as some people from Italy, Russia, Poland, and even Germany who have olive skin. I have seen quite dark-featured Russians! While it's true there are very white Spanish and Italians and even Greek people, my theory is that there has to have been a dark-skinned native population in these countries. I'm thinking the old Europeans were invaded by the Gothic peoples and hence the variety of blond and dark looking Spanish, Italian, Greek, etc. But I didn't know Poland also had this. Meanwhile English and Scottish tend to look entirely northern European, but even there there's a few exception (Victoria Beckham, Catherine Zeta Jones, Sean Connery). I don't know if it's invasions by Arabs, but I don't think so because people from Italy and Spain don't look Arabic.