Perfectly legal
In most countries in Europe, and even worldwide, knife throwing is perfectly legal.
Of course, many countries have rules about carrying knives in public, and transporting them, but those rules are so open to language interpretation that it's best to use local sources to learn about them.
A good collection of international weapon laws on knives is maintained at Wikipedia.
Restrictions
However, there are some countries where EuroThrowers members have reported restrictions on knife throwing. The throwing itself is not banned, but sometimes throwing knives are illegal. People throwing in those countries use ordinary (but blunted) knives for their sport.
Denmark
A EuroThrowers member succeeded in having the ban on knife throwing lifted. You now can obtain permission to throw knives in private and in public.
Finland
A member reports that throwing knives are legal as of 2003!
Hungary
Throwing knives is reported to be legal again.
Switzerland
Only asymmetrical throwing knives are legal, but must nevetheless be transported in a locked container. Carry a paper explaining to which sports event you are traveling.
Asymmetrical throwing knives are those where one side of the blade looks completely differen than the other side - as for example with the Faka throwing knife.
Symmetrical throwing knives - which are considered weapons and need special permissions - have a mirrored look on both sides of the blade (no matter that one side is not sharp). Exmples for these not-even-transport throwing knives are the Viper or Stinger/Arrow models.
Details see Gesetzliches zu Wurfmessern in der Schweiz.
The Schweizer Werfer Vereinigung has succesfully lobbied to get the inital ban on asymmetrical throwing knives lifted.
Australia
Importing throwing knives requires a police permit (according to the customs). If throwing knives are then legal seems to vary depending on the jurisdiction region within Australia.
Netherlands
Throwing knives are illegal.
Belgium
Throwing knives are illegal. (Article 3, §1 of the 2006 Weapons Act)
Luxembourg
Throwing knives are illegal. (Loi du 15 mars 1983 sur les armes et munitions", §1 c)