Higgs field was theoretically predicted in order to explain from where massive particles gain their mass. According to the formula, E=mc2 mass of the given particle has the origin in its energy. Mass-energy equivalence principle does not require the existence of the particular field which would carry particles mass. The Standard Model explains that Higgs field has appeared after Big Bang when the universe starts cooling down. Several pieces of research confirm Big Bang never happen which represents the serious insufficiency of the Higgs mechanism. The next insufficiency of the Higgs mechanism is that the gluons which represent 99% of proton mass do not interact with the Higgs field. In CERN experiment are colliding about hundred millions of pairs of protons each second and only a few collisions produce Higgs boson. What about other pairs collisions? Why only 1 in ten billion of collisions produce Higgs boson? This special issue is dedicated to stress out weaknesses of Higgs mechanism and to give alternative models for the origin of mass of elementary particles.