Are fields (electric or magnetic) real? Are fields (electric or magnetic) real or they are just theoretical concepts to explain lot of stuff which couldn't be explained by Newtonian action at a distance approach?
 A: It's hard to define "real" here, because we humans can't feel it.  Or can we?  The hairs on your head will stand up in a powerful electric field.  That seems real enough.  And a magnetic field is real in the sense that magnetic and magnetically-permeable objects that enter it are affected by the field. So are very real charged bits from the sun, so we get auroras.  There's evidence that birds feel the effect of magnetic fields as well.
It was Faraday who defined the concept of a field.  Before that, they just knew that magnets attracted each other.  But we know now that the field has a defined shape, direction, and strength, and that real objects interact with it.
That seems pretty real to me. 
A: When by "real" you mean that in a scientific sense they exist as actual things as opposed to just imagined or supposed then they are real because they can be measured by independent observers under the same conditions obtaining the same results. There are mathematical entities, however, like the electromagnetic vector potential or the Schroedinger wavefunction, whose physical reality is often negated or disputed. 
A: It is not necessary to see something to accept it as real, but it's necessary to be able to measure it. We can't see an electric current flowing, but we can measure it. Measurement not has to be quantitative only, it's enough to see, hear, smell etc. an influence of the phenomenon on another object.
An other question is the reality of field lines as well as of the constituents of electric and magnetic fields. Until now there isn't any measurable phenomenon of quantization of such fields. So this fields could be treated only as statistical quantities. For example the current commonly is a statistical quantity as long as one not calculated the amount of the flowing electrons. For electric and magnetic fields the constituents are unknown and any investigation about them seems not to be necessary.
