Odoo Subscriptions allows to manage subscriptions and payments for subscription-based (recurring) businesses. As of Odoo 9, there is no other choice than to process the automated debit/credit card charges through the Ingenico Payment Services gateway (also known as Ogone). This gateway allows to connect to the following processors. Contact your bank to make sure it can receive payments from one of these processors. You will then have to open an account with that processor and with Ingenico.

The information flows as follows:

Odoo ↔ Ingenico ↔ Processor → Your bank

=== EDIT Odoo 10 ===
As of Odoo 10, Odoo Subscriptions also works with the Authorize.net gateway.
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=== EDIT Odoo 12 ===
As of Odoo 12, Odoo Subscriptions also works with Stripe.
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Payment processors compatible with Ingenico Payment Services:

• Agricultural Bank of China (ABC)

• ANZ Bank

• Atos Worldline - Banksys

• Atos Worldline - BCC

• B+S Card Service

• Bangkok Bank (BBL)

• Bank of China International (BOCI)

• Barclaycard UK

• Barclays UK (Mult. cur.)

• CASHU

• Chase Paymentech

• ConCardis

• Deutsche Bahn

• Diners/Citibank Card Acceptance

• Diners/Citibank Card Acceptance BE

• DTAUS

• Elavon (BE)

• Elavon (INT)

• Euroline (SEB)

• European Merchant Services B.V.

• EVO Payments International GmbH

• FirstData US

• Global Payments (HSBC International)

• Kasikorn Bank (KBANK)

• PAYPAL

• SIX Payment Services (Europe) S.A.

• SIX Payment Services Ltd

• UATP

Payment processors compatible with Authorize.net (source)

North American Processor Connections:

• Chase Paymentech - Tampa Processing Platform (USD, CAD)
• Elavon (USD, CAD)
• First Data Merchant Services (FDMS)
  • First Data Omaha (USD, CAD)
  • First Data Nashville (USD, CAD)
• Global Payments - East Processing Platform (USD, CAD)
• Heartland Payment Systems (USD)
• TSYS Acquiring Solutions (USD)
• Heritage WorldPay (USD)
• Vantiv / WorldPay Core (USD)
• EVOSnap (USD, CAD)
• NAB EPX (USD, CAD)

• FdiAus (USD, AUD, NZD)
• WestPac (AUD)

• AIBMS (USD, GBP, EURO)
• AmexDirect (USD, GBP, EURO)
• Barclays (GBP, EURO)
• Cardnet (GBP)
• HBOS (GBP)
• HSBC (USD, GBP, EURO)
• Streamline (USD, GBP, EURO, AUD)
• EVOSnap (USD, EURO, GBP, CHF, DKK, NOK, PLN, SEK)