Rinse the kumquats. Juice them on a citrus juicer or squeezer. It's OK if the peels rip, as long as they aren't shredded into tiny pieces. Set aside the peels to be used for candied kumquat peels and kumquat syrup.
Remove any segments of fruit and dangling strings from the peels. Try to keep the peels in relatively large pieces.
Use the syrup for mixed drinks, cake glaze, or as a topping in place of maple syrup or honey.