Crimes in other countries?

say you commit a crime in a foregin country, start with Assult, rape, robbery, murder, ect. and you get back to the us before your arrested or anything, what happens if you are then a suspect, do you go to jail in the us, does the us or foregin countries embassy (empoloyees?) embassadors come get you and take you to the country where the crime was commited, and visa versa if you do a crime here a flee the country what all can go down?