I have a Polish student, which makes a change from the usual Spanish/Catalan speakers. As is common with speakers of Slavonic languages she is unsure about articles. We went through the units in English Grammar in Use and she got the general idea but was confused by the more arcane aspects of article-usage. She could accept the post-definition use, even that Cambridge University and the University of Cambridge are two equally valid names for the same institution, but could not understand why we have Canterbury cathedral and Euston station but the National Theatre and the BBC; or Mount Everest but the river Thames. She could understand that the United Kingdom and the United States have articles because they have countable nouns in their names, and that the Canaries and the Philippines have the implied word Islands, but why, she asked, is it the Vatican? I regret to admit that I did not have a convincing answer for her except to say that there are always exceptions and peculiarities in language.
This map from Wikipedia shows the form and distribution of articles in and around Europe.
I have touched on a problem with articles in a previous post.