If you hold bonds through a brokerage, they'll likely automatically be cashed in when they mature, or stop paying interest. But otherwise, as is often the case with U.S. savings bonds, you're simply giving the bond issuer a no-interest loan by not cashing them in, since they are no longer paying out any interest.
Bonds that have matured do not accrue any more interest. If you don't cash the bond in, you're allowing the U.S. Department of Treasury to hold your money interest-free.
A bond is essentially a loan to a company or an arm of the government, paid back with interest over a period of time. After that time is done, usually after years or decades, the loan is paid back and the bond is said to mature.
While today it's possible to have bonds automatically make payments electronically, historically it was necessary to contact whoever issued the bond and request payment. If you fail to do so, you are essentially giving whoever issued the bond a free loan, since they are holding your money and no longer paying you any interest on it.
U.S. Savings Bonds
This is often common with U.S. savings bonds, which many people receive as gifts as children and forget that they still have. If you have a savings bond, you can contact the U.S. Department of the Treasury to see how much it is worth and if it is still earning interest. If not, you can work with a bank or the treasury department to cash it in. Remember that you'll owe tax on the interest that you've earned on the bond. If it is still earning interest, you may be able to claim some of that interest early to avoid being taxed on it all in one year, which may be advantageous depending on your overall tax situation.
If you're not sure if you have any savings bonds, you can also fill out a form asking the treasury department to check on your behalf, usually using your Social Security number. If one is found, the treasury department can help you reclaim it and, if you wish, cash it in.
Traditionally, many organizations issued bonds known as "bearer bonds," payable to the person who held a physical certificate. For a variety of reasons, those have generally fallen out of favor, and most people now own bonds through a brokerage or other financial institution.
But if you do have any old bearer bond certificates, it can be worth checking if they are earning interest or if you wish to cash them in. Do this by contacting the organization that issued the bond or working with a brokerage or another expert.
Even if the company that issued the bond has gone bust, the certificate itself might still have value as an antique.