Vizing conjectured in 1963 that the domination number of the Cartesian product of two graphs is at least the product of their domination numbers; this remains one of the biggest open problems in the study of domination in graphs. Several partial results have been proven, but the conjecture has yet to be proven in general. The purpose of this thesis was to study Vizing's conjecture, related results, and open problems related to the conjecture. We give a survey of classes of graphs that are known to satisfy the conjecture, and of Vizing-like inequalities and conjectures for different types of domination and graph products. We also give an improvement of the Clark-Suen inequality. Some partial results about fair domination are presented, and we summarize some open problems related to Vizing's conjecture.