My organization is looking to set up a monthly giving program -- does CiviCRM have the capacity to process/store recurring donations?
Civi works with a number of payment processors, some of which do recurring payments.
I think the best list of which do what is here: https://wiki.civicrm.org/confluence/display/CRMDOC/Payment+Processors
And on top of that, for European organizations that need it there is an extension CiviSepa to work with SEPA recurring contributions. But not sure what region you are in :-)