Federal Reserve Notes are printed by the Bureau of Engraving and Printing at facilities in Washington and Fort Worth.
They're then distributed through the Federal Reserve banks located in 12 different cities around the country. Each bank is responsible for requesting currency and transferring it to the private banking system.
Contrary to popular misunderstandings, the US Mint doesn't have anything to do with paper money. The Mint only issues coins.
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The phrase is actually "Redeemable in Gold" rather than "Backed by Gold". Federal Reserve notes were and remain so-called fiat money, i.e. paper currency that represents an obligation of the central banking system that is not dependent on precious metal. The phrase was removed from Federal Reserve notes starting with the 1934 series, issued the year after Franklin Roosevelt took the US off the gold standard. It had appeared on previous Federal Reserve notes beginning with the first series in 1914. Strictly interpreted, the full phrase limited gold redemption only to those bills presented at the Treasury itself or any of the twelve Federal Reserve banks, places that the average person would rarely visit. The clause "... or lawful money" was never clarified but was interpreted to mean any coin or bill issued by the government, including other Federal Reserve notes.
The last president on US $10 bills was Andrew Jackson on Series 1914 Federal Reserve Notes. Since Series 1928, the only person on the US $10 Federal Reserve notes, despite several redesigns, has been Alexander Hamilton, the first Secretary of the Treasury.
Who or what group of individuals formed the Federal Reserve
Until 1991 all Federal Reserve Notes were printed in Washington. Please post a new question with the bill's denomination and what letter, if any, is next to the date. You don't have to copy out the serial number, though.
They're identified right on the bill - the signatures are those of the Secretary of the Treasury and the Treasurer of the United States.