Lots of talk of trains and trams to Alicante and Benidorm but at the moment it is not possible to get direct public transport to the airport from anywhere south of the airport eg Torrevieja, Orihuela Costa where so many expats live. There is a good bus service between Cartagenia and Alicante but it doesn't stop at the airport. There is no obvious reason why there should not be a service from Torrevieja to Alicante Airport. It would be well used but perhaps the bus companies are owned by taxi drivers.
