My thesis investigates the S-packing coloring of subcubic and non-subcubic graphs. The S-packing coloring of a graph G, where S = (s1, s2,… ., sk) is a non-decreasing sequence of positive integers, involves assigning colors to the vertices of G such that any two vertices assigned the same color si are at a distance of at least si + 1 in G. S-packing coloring is a fundamental problem in graph theory. It is a mix of proper coloring and its extension distance coloring. This topic was initially introduced to address broadcast frequency assignment problems, aiming to minimize interference between signals. Such practical applications highlight the relevance of this concept in solving real-world challenges. My research aims to address several open problems in this area, with a key focus on the conjecture: "Is every subcubic graph, except the Petersen graph, (1,1,2,2)-packing colorable?"