There are five Bourne movies in total, released between 2002 and 2016. The correct order to watch them is:

1. The Bourne Identity (2002)
2. The Bourne Supremacy (2004)
3. The Bourne Ultimatum (2007)
4. The Bourne Legacy (2012)
5. Jason Bourne (2016)

This order is both chronological and the order in which the films were released. It is important to note that the 1988 made-for-TV movie, The Bourne Identity, is not part of the franchise and has nothing to do with the films listed above.