All US taxpayers living overseas have to file a federal income tax return each year if they earn over $10,000, or just $400 of self-employment income, regardless of where they live or where their income is generated. They may also have to pay US income tax on their foreign earned income, depending on their circumstances.
If you live overseas and you are eligible to receive social security benefits, you may receive them abroad (unless you live in a country under US sanctions, such as North Korea).
Regarding paying US social security and Medicare tax while living abroad, if you are working for a US employer, or if you are living abroad and you're self-employed, you are still required to pay US social security tax.
If you're living abroad and working for a foreign employer on the other hand, you are not required to pay US social security ( depending on the country, unless you are just working abroad on a short term contract).
The US has what are called Totalization agreements currently with 25 other countries, namely:
Australia, Austria, Belgium, Canada, Chile, Czech Republic, Denmark, Finland, France, Germany, Greece, Ireland, Italy, Japan, Luxembourg, Netherlands, Norway, Poland, Portugal, Slovak Republic, South Korea, Spain, Sweden, Switzerland, the United Kingdom.
Totalization agreements grant US citizens working and paying social security tax in these countries the ability to use the social security tax that they pay in these countries towards qualifying for US social security benefits, without affecting the ability or right to claim social security benefits in the other country based on the contributions made there.
“Your benefits can be mailed to your foreign address or deposited electronically in a financial institution located in your country of residence or in the U.S.” - Chad Creveling, Wall Street Journal
Furthermore, US tax-payers' non-US citizen spouses who are citizens or residents of these countries are eligible for US spousal and survivor benefits.
Social Security tax rules for expats can be a little complex, as they vary depending on the expat's particular employment circumstances as well as which country they are living in. If you have any doubts regarding your particular situation, don't hesitate to consult an expat tax specialist.