Formal:

Auf Wiedersehen!

When you say farewell or goodbye in a formal way, you can also shake hands, just as you might when you greet someone.

Depending on the context, if you want to be very polite you can also say Vielen Dank (thank you). For example:

Vielen Dank für Ihre Hilfe – when someone has helped you

Vielen Dank für den netten Abend – when you've had a nice evening together

Informal:

Tschüss!

This is the most common way of saying goodbye. These days it is used throughout Germany.

Most people say goodbye informally in various ways:

Ciao! – especially popular among young people

Bis bald! – when you will see someone again soon

Wir sehen uns! – what friends say to each other when they see each other regularly

Mach's gut! – a relaxed way of saying goodbye when you want to wish the other person luck

Special cases:

Einen schönen Tag noch! – a polite goodbye, especially used in shops

Auf Wiederhören! – a formal goodbye on the telephone

Regional variations:

Servus! – In southern Germany and Austria, it's not only used to say "hello", but also to say "goodbye"

Tschö! – in the Rhineland in western Germany