For most countries that I'm aware of, when you apply for permanent residency you need to submit a criminal background check from your home country. They don't want to let in fugitives or violent criminals.

For US citizens, that means the FBI (although you generally have your fingerprints taken by your local police, and send the fingerprint card to the FBI). Since I was already in South America, I had my fingerprints taken at the Paraguayan Interpol office. I sent them to the FBI in the US and waited about 75 days to get my results back.

I also had to get fingerprinted for a criminal background check by the Paraguayan police, even though I had never been in Paraguay before.

Other than that, the process was fairly simple - a copy of my birth certificate (certified by the Paraguayan consulate in the US), a couple of visa-sized photos, and US$5000 to deposit in a local bank for the duration of my application (to prove I'm financially solvent). If you're married or divorced, you'll also need to have those papers legalized.