This work is devoted mainly to General Relativity particle dynamics in the whole range of gravitational field strength in the GR academic framework. There is an opinion among the Physical community that the problem is fully studied and understood in all its mathematical and physical aspects, however, it is not true. The exact full solutions of the GR equation of particle motion in the whole range of fields have never been studied. It is shown that approximate solutions are not sufficient for understanding the problems at the deeper level and formulating the criterion of empirical testing of the Einstein's theory. As an example of the weak field problem, for the first time we study Einstein's Mercury perihelion advance in full details including exact solutions in terms of angular and temporal dynamic variables. Similarly, exact solutions were obtained for strong field problems. Both studies reveal some controversial issues, actually, known in literature but not paid sufficient attention, or remained physically unexplained, for example, a nonremovable singularity, the superluminal motion, solutions in terms of two dynamic temporal variables $t$ and $\tau$, and other issues. Consequently, we conduct an additional study of the particle dynamics in the alternative, the Special Relativity framework, with the conclusion that the GR controversial issues do not arise there. Overall, novel methodological and technical ideas and the corresponding results related to the whole range of field strength are claimed. The work is intended to stimulate further constructive discussions among researchers in Fundamental and Modern Physics, and Physics Frontiers, also, it has a pedagogical value.