When I went to school I was specifically told that fields are material (they occupy some region in space, and they "exist" there) and continuous.
Recently, studying quantum physics I came across the notion of gauge particles which carry all kinds of interactions. From this perspective, if I judge it correctly, fields do not exist - they are essentially these particles being exchanged all the time between the interacting objects.
So, what is the current approach to the concept of fields?
