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Standard Customer Guidelines.
These apply for flights which BA have cancelled - typically you will get a text or email showing this, or a message showing on Manage My Booking.

Options open to you:
1) Refund - note if you have partly taken the flight then this will be a partial refund. If it's a full cancellation you should get back the direct costs, but if you already moved the flight for other reasons previously you may lose some components.
2) Rebooking - on alternative BA or IB flights within 3 days before or 14 days after - you just need space in the cabin. This also works for Joint business routes over the Atlantic and Siberia
3) Rebooking on BA or IB but outside the dates above - you need space in the fare bucket - ditto for AA and AY on Joint Business routes.
4) Rebooking on QR joint business routes - but this needs to be between 3 days before or after, you need space in the cabin.
5) Go to another gateway within 300 miles - so you can travel to Verona instead of Venice, for example. 3 days before or 14 days after rule applies

If the cancellation is for flights within 24 hours, BA Contact Centres can rebook to other IATA airlines that BA has a ticketing agreement with - that's a long list, but it includes most household names as well as oneworld airlines. It does not include easyJet, Ryanair or Jet2.

If you booked via a travel agents or an online travel agents (OTA) such as Expedia you will need to recover your refund via the agency, not via BA. However if you are within 24 hours of the first sector - or thereafter - you can ring BA on rebooking options.