How to pay taxes when living nomadically

If I transition to a nomadic lifestyle, how does that affect my taxes?  Specifically, would I need to file a state tax return in any state that I'm physically located in when earning money in that state (even if for just a short time, e.g., a week or a month)?   Or do I need to maintain an official "permanent" address that would be associated with my state taxes (I currently live in MO)?   Does a PO box count as an official permanent address?  Additionally, if I'm working/living outside of the US, how does that affect the way I'd need to pay my US taxes?