This paper reviews major approaches to the description of subatomic particles, suchas leptons, quarks, hadrons and nucleons. Among these approaches are quantum chromodynamics, soliton models, bag models and others. The main accent is on a theory of thereal physical space that acts as a scene on which all high-energy events take place. Wediscuss how a lepton and quark appear in the space constituted as a tessellation lattice ofprimary topological balls — the only structure that mathematics (i.e. set theory, topologyand fractal geometry) offers to the constitution of ordinary physical space. Since leptonsand quarks emerge in the tessellattice from a topological ball, they must interact withthis substrate. The principles of the interaction of subatomic particles with space andthrough space between themselves are considered in detail. The approach: i) states thatreal quarks possess the integer charge ±e and they periodically change to the monopolestate (hence, canonical particles are dynamic dyons); ii) naturally solves the problem ofconfinement of quarks; iii) reveals the dynamics of quarks in hadrons; iv) discloses an innerstructure of the proton and neutron; v) calculates the radius of the proton; and vi) derivesthe nuclear forces as the result of both direct coalescence of surfaces of the nucleons andthe overlapping of spatial excitations (named inertons) generated by the nucleons at theirmotion through the tessellattice. Experimental results showing nuclear transformationsin samples affected by artificially generated inerton fields are demonstrated.