electromagnetism - Is the concept of a field necessary to electrodynamics?

I've read (in Griffith's text) that it is "possible, though cumbersome" to dispense with the field concept in electrodynamics entirely and instead use an action-at-a-distance theory.

What exactly is meant here? Since fields physically exist, why is this permissible?

Edit: In light of the answers below, I'm curious about the apparent contradiction here - if fields physically exist, then how can one dispense with the field concept?