If photons have also particle properties why should they not collide with each other? If photons have also particle properties why should they not collide with each other? Collisions between fermions are possible as collisions between fermions and photons(bosons) except collisions between photons (that are described by Albert Einstein as particles in the photoelectric effect experiment). Why?
 A: The framework of photons is the elementary particle quantum mechanical frame. Photons are point particles in the Quantum Field Theory,  QFT, of the standard model of particle physics. Particles do not "collide",but scatter  off each other, and the probability of the interaction happening can be calculated by QFT. Point particles because the calculations depend on Feynman diagrams, which  assume that the  interactions happen at a point, called a vertex. Here is  the Feynman diagram of  photon photon interactions to lowest order


Feynman diagram (box diagram) for photon–photon scattering, one photon scatters from the transient vacuum charge fluctuations of the other

Each point vertex in the calculation carries a small number, the coupling constant, which for electromagnetic interactions is 1/137, and thus the final number of photon-photon scattering is very very small. That is why "collisions between photons" need special experiments with high accuracy in order to be detected.
It is one of the reasons that classical electrodynamics is still very successful in describing interactions of classical electromagnetic waves with matter.
The above is true when the  energy of the photons is low. For energies of the order of MeV , special relativity allows the production of pairs of particles and QFT predicts the crossection of the interaction . This  has been seen experimentally. There are even plans to build a gamma-gamma collider to study further the full gamut of elementary particle interactions.
A: Hans Heinrich Euler was the first who asked your question in his great PhD thesis "Über die Streuung von Licht an Licht nach der Diracschen Theorie (On the scattering of light by light based on Dirac's theory) in 1935." Today, we know the answer.
In fact, Photons can interact with each other. The Standard Model of elementary particles predicts the following (experimentally detected) results for the interaction of photons.:

*

*the existence of photon-photon scattering (by the well-known box diagram in QED). Also known as Delbrück or light-by-light scattering.

*the production of some massive particle-antiparticle pairs from colliding real (on-shell) photons. For example, $\gamma\gamma\to p\bar p$ reaction.

The photon-photon scattering have been observed recently in ATLAS [1]. This process can be understood via the following box (second order) Feynman diagram

which is the lowest possible order at which photons scatter. Since it is not at tree level, the photon-photon (light-by-light) scattering is extremely rare (In the corresponding Feynman amplitude, there exist four massive fermion propagators).
On the other hand, photons can interact with each other and, e.g., proton-antiproton pairs have been observed from colliding real (on-shell) photons ($\gamma\gamma\to p\bar p$ reaction) [2,3]. For more details and the relevant Feynman diagrams, see my answer in this PSE link: Is it possible to create protons from photons?

References
[1] ATLAS Collaboration., Evidence for light-by-light scattering in heavy-ion collisions with the ATLAS detector at the LHC, Nature Phys 13, 852–858 (2017).
[2] L3 Collaboration, "Proton–antiproton pair production in two-photon collisions at LEP", Phys. Lett. B 571, 11 (2003), arXiv:hep-ex/0306017.
[3] M. Kłusek-Gawenda, P. Lebiedowicz, O. Nachtmann, and A. Szczurek, "From the $\gamma\gamma\to p\bar p$ reaction to the production of $p\bar p$ pairs in ultraperipheral ultrarelativistic heavy-ion collisions at the LHC", Phys. Rev. D 96, 094029 (2017), arXiv:1708.09836.
A: The QED Lagrangian is:
$$ L = \bar\psi[i\gamma^{\mu}(\partial_{\mu}+ieA_{\mu})-m_e]\psi-\frac1 4 F_{\mu\nu}F^{\mu\nu} $$
The $ie(\bar\psi \gamma^{\mu}A_{\mu}\bar\psi)$ terms shows us that photons ($A_{\mu}$) couple to spinor ($\gamma^{\mu}$) electrons ($\psi$) with a strength proportional to charge ($e$).
The portion due solely to photons is:
$$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+gA_{\mu}A_{\nu}$$
The $\partial_{\mu} A_{\nu}$ terms are due to the electric and magnetic fields, while the photon-photon coupling term is:
$$ gA_{\mu}A_{\nu} $$
The problem with it is as follows:
$$ g = 0 $$
Photons have no charge, so they don't couple to each other.
Of course, there are the light-by-light scattering diagrams:

which means two photons can interact, but it is indirectly, as they are scattering off a charged quantum-vacuum fluctuations.
A: Having particle properties does not mean that a photon is a small sphere traveling at the speed of light. Collisions in quantum mechanics mean that particles interact and modify their free-particle behavior.
Quantum electrodynamics predicts an electron-electron interaction, electron-photon interaction, but also a photon-photon interaction. However, it is weaker than the electron-electron interaction and not observable in practice at usual energies and energy densities. Experimentally it is studied at high energies ( see, for instance, the Wikipedia page on  two-photon physics ))
A: If photons are neutral particles with a very small size, the interaction may be very improbable (small cross section). But there is a more long-distance interaction of photons - they are bosons and thus somewhat "attract" each other, roughly speaking. Lazers are typical devices using this effect.
