What is the quickest way to checkmate?

Strictly speaking, a checkmate is possible on black's second move, with the following turns:

1. f3 e5 2. g4 Qh4#

This mate is so unlikely that it would need cooperation between the two players. The quickest mate that actually happens in Chess (usually at the beginner level) is:

1. e4 e5 2. Bc5 Nc6 3. Qh5 Nf6 4. Qxf7#