In this talk, we consider the problem of hyperball (hypersphere) packings in n-dimensional hyperbolic space.  We first review previous results on hypersphere packings and coverings. Then, we prove that to each n-dimensional congruent saturated hyperball packing, there is an algorithm to get a decomposition of n-dimensional hyperbolic space into truncated simplices. We prove, using the above results and the results of  Miyamoto, that the density upper bound of the saturated congruent hyperball packings related to the corresponding truncated tetrahedron cells is realized belonging to a regular truncated simplex and in 4-dimensional hyperbolic space we determined this upper bound density ~ 0.75864. Moreover, we deny A. Przeworski's conjecture regarding the monotonization of the density function in the 4-dimensional hyperbolic space. (Partially joint work with Arnasli Yahya.)