What is invisible to x-rays? Is there any physical matter which won't appear on x-ray i.e. invisible matter?
I think I learnt at university that all phsyical matter appears on x-ray and there is no matter that can be invisible. Is this true? For instance, does dark matter and antimatter appear on x-ray or what technology is used to prove that there is something instead of nothing?
 A: While aiming to take a X-ray picture of a bone fracture, soft tissue all around the bone becomes invisible or unclear. In this case soft tissue is invisible or unclear.
A: There do not exist materials made of antimatter, so even though they would behave completely symmetrically to the corresponding matter materials, the fact is irrelevant.
Dark matter reacts only with gravity, and X-rays are electromagnetic waves. To all intents and purposes, as far as possibility of measurements, dark matter is transparent to X-rays, since the gravitational interaction of X-rays is miniscule.
What is your definition of transparency? For example flesh is transparent while bones and metal are not for an X-ray photograph. 

Hard X-rays can penetrate some solids and liquids, and all uncompressed gases, and their most common use is to image the inside of objects in diagnostic radiography and crystallography.

There will always be some interaction of X-rays through matter, transparency has to be defined.
A: The question can be rephrased as "is there any form of matter that does not absorb X-rays?", because objects only appear on an X-ray plate if they've absorbed some of the radiation.
Anything with electrons in it, i.e. all normal matter, will absorb X-rays to some extent. I think the interaction cross-section for photons and dark matter is effectively zero (I wouldn't swear it was exactly zero) so you re correct that dark matter wouldn't absorb X-rays and wouldn't appear on an X-ray plate.
This is probably getting away from the spirit of your question, but you may have heard of metamaterials. You can design metamaterials that (in some very tightly controlled circumstances) don't absorb longer wavelength light like infra-red. In principle it might be possible to do something similar with X-rays, but at present our technology is miles away from this. Metamaterials work by diffracting and/or refracting light so they need structures that are around a wavelength of light in size. X-ray wavelengths are down around the 1 Angstrom mark and this is far too small for us to engineer at present.
A: X-rays will interact strongly with anything electrically charged. They interact most strongly with dense materials with high-Z nuclei since they have a lot of strongly charged nuclei to scatter off of. This is why on an x-ray image you effectively see a projection of the density of an object. All normal materials are made of atoms and so will interact to some degree with x-rays, although low density materials can be effectively transparent to them.
Electrons will also interact with x-rays since they are electrically charged.  Uncharged particles such as neutrons will interact much more weakly with x-rays since the x-ray photons have to get inside the neutron rather than just close to it. 
Hypothetical materials made from antimatter would interact with x-rays in exactly the same way as normal materials. More interestingly, when anti-electrons annihilate with normal electrons they produce x-rays with a distinctive energy that are easily identified. This is the typical way of detecting positrons (anti-electrons). 
Dark matter certainly interacts gravitationally and does not interact electromagnetically, but that is not the only possible interaction. If dark matter can interact non-gravitationally with normal matter there may be very rare (high order) interactions with photons such as x-rays. In any case, interactions between dark matter and x-rays are far too faint to be detected. 
A: Beryllium but it's toxic. It's transparent 
