This article was written by Daniel Schlichtholz, the developer of MySQLDumper - a free and useful script for backing up and restoring databases. The original language was German and I have translated it to English, because there are people using umlauts and other special characters in other languages who would benefit from reading this article.
The original article, in German, can be found here.

So, here it goes.

The Umlaut-Problem: