Is a routing number always required to transfer money?

In the United States, yes, you do need the ABA number or the routing number in order to make the transfer. This is a 9-digit number that along with the bank account number makes the interbank transfer possible.

If the transfer is intra-bank (within the same bank), then the routing number may not necessarily be needed.

For cross-border transactions, if the transaction is inbound into the United States, a routing number would be essential information.

If the transaction is out-bound, then the routing number is not needed.

For transactions that settle outside the US, or even originate and settle outside the United States, it depends on what sort of a transfer is being done. In SEPA there is no concept of a routing number. IBANs are used. Other ancillary information used other than IBANs are SWIFT and BIC codes.