Odoo is an open source accounting software. With much more features than only accounting, such as warehouse management, project management, HR management etc.
CiviCRM is an open source software for non-profit and is great in membership management, donor management etc.
For one of our clients we have been working in integrating both CiviCRM and Odoo where Odoo is leading in the financial data and CiviCRM is leading in the contact and membership data.
Meaning that in CiviCRM we create memberships and memberships contributions. Those contributions are synced to Odoo as invoices. In Odoo the invoices marked as direct debits are exported and send to the bank and in Odoo we read and reconcile the bank statements and marking the invoices as paid. As soon as an invoice is marked as paid the contribution in CiviCRM is also marked as completed. In addition when an invoice is created in Odoo this invoice is also created as a contribution in CiviCRM.
The financial department could also remind people of unpaid contribution from within Odoo. This comminication with the members is logged as an activity in CiviCRM. So the membership department could see that the financial department has contacted this member about unpaid contributions.
At this moment the synchronisation between Odoo and CiviCRM is in a testing phase.
Interested in hearing more about this work. The accounting piece (which I have limited exposure) is a much desired piece of the puzzle for the ORGs that I work with. They want to be able to "balance" the books and right now this is a challenge for them, either because of their work flows or how they are exporting the data and importing into their accounting. Let's exchange contact details.
It sounds like your organisations are looking for Odoo. As Odoo is more an financial accounting system. You could find Odoo at http://www.odoo.com it is open source
Odoo is overkill for us after looking at it. We currently use Aplos for our accounting (http://www.aplos.com). They have recently implemented an api and we are looking into doing some syncing between Civi and Aplos.
Our thoughts were to have a scheduled job that would run daily and create a Civi accounting batch for the previous day. That batch would include all financial transactions that civiCRM processed that would be deposited into the bank account.
That total would then be posted to Aplos as a Deposit transaction with the Civi Accounting batch ID located in the comments of the transaction. That way if there was ever any question we could cross reference between the two.
Also, if a refund is processed through Civi then that would be posted as a Debit to Aplos with the Payee just being REFUND and in the comments it would notate the details.