Is it correct to say that electromagnetic waves does not require a medium? I can conceive of a particle existing in empty spacetime, but not a wave. A wave appears to me at least, to insist upon a medium for its very definition. 
I understand that the 19C physicists postulated the lumineferous aether for this very reason, but abandoned it in favour of the electromagnetic field.
In the textbooks I've looked at, its always said that the physicists eventually understood that a wave does not necessarily require a medium. But it seems to me tht they abandoned the idea of a medium mechanically conceived (I assume because of the high prestige of newtonian mechanics, its status as a defining, fundamental paradigm which privileges our naive & intuitive understanding of mechanical phenonema) for a medium conceived in a a more general manner.
Afterall, this field permeates spacetime, and its state evolves through time. This is how I would naively conceive of a medium in a general sense.
I should add this is maybe more of a history of physics question, and I've added a tag as such to stop people getting upset.
 A: In classical electromagnetism, once you pick a reference frame, the electric and magnetic fields have a value at each point of spacetime. The time-evolution of these fields obeys Maxwell's equations. That's about all there is to say on the matter; I doubt further philosophizing will get you anywhere. 
You can say "there is a medium" or "there is not a medium" and it doesn't change anything unless you give some more detail about what that means. It sounds to me like you want to say "there is a medium", but all the physics will still be exactly the same as it is for people who say "there is not a medium". In that case it is a moot point. Mystical undefined medium or no, radio telescopes will work the same. On the other hand, aether, as it was conceived in the late 19th century, made concrete predictions about things like experiments with interference, and those predictions were wrong.
You can consider more modern physics than classical electromagnetism, thinking about EM radiation either in general relativity or as a quantum phenomenon. If you do this, perhaps you will find an analogy that you find more appealing. But ultimately physics is a mathematical description of nature, not a particular, anthropomorphic interpretation. When the action is large compared to Planck's constant and when distances are large compared to the Bohr radius divided by the fine structure constant, classical electromagnetism works really well, and I don't see the need to introduce extra concepts into it. If it makes you feel better to describe the existence of electromagnetic fields as "being a medium", so be it. For one, I can do without.
A: What is colloquially called ''empty space'' isn't really empty - it is filled by the electromagnetic field and the gravitational field; it is called empty only because it doesn't contain (nonzero) matter fields.
The electromagnetic is the medium that carries electromagnetic waves, such as the air density field (colloquially just called ''air'') carries sound waves and the water density field (colloquially just called ''water'') carries water waves. 
Indeed, electromagnetic waves are nothing else than propagating high-frequency oscillations in the electric fields, in precisely the same way as sound waves are propagating ohigh-frequency scillations in the pressure field of air (or any other mechanical medium), and water waves are propagating low frequency oscillations in the mass density field of water.
A: By using Maxwell's equations it can be shown that electricity and magnetism should somehow be able to possess wave behavior. It even predicts the velocity of electromagnetic waves (and therefore naturally, it should lead to a medium for electromagnetic wave to propagate in, or is it?) which is a constant number. But the problem exactly is this! With respect to what frame of reference that velocity is a constant? (The expression for the velocity of light only contains permittivity and permeability of space which are both constants).
Ok, what happens if we assume a medium?
If we assume that medium ('aether') to be an inertial reference frame we immediately run into trouble, because our expression predicts a constant velocity. In theory, the speed of light ('C') is a constant, and so is in experiment (try measuring C in laboratories moving in different velocities at the same instant, you'll always come up with the same value for C). If we continue to believe in the existence of aether we'll therefore have to assume that aether is somehow always at rest with respect to our laboratory no matter how fast it moves, which is totally unfair! So is aether a non-inertial frame of reference? No! Measure C in an inertial frame, it always stay constant!
So if we assume the existence of such medium any longer we'll have to admit that it cannot be regarded as either an inertial frame or a non-inertial frame of reference. Something we assumed must be wrong!
