Let D be a Dedekind domain. D is a half factorial domain (HFD) if for any irreducible elementsof D the equalityimplies that s = t. D is a congruence half factorial domain (CHFD) of order r>1 if the same equality implies that s = t (mod r). In this paper we expand upon many of the known results for HFDs and CHFDs (see [6] and [7]) as well as introduce the following new class of domains: if k≥1 is a positive integer then D is a k—ha1f factorial domain (k—HFD) if s Skin the previous equality implies that s = t. In section I we explore the interrelationship of the HFD, CHFD, and k—HFD properties and offer a method for constructing examples of k—HFDs and CHFDs by viewing the class group of the given domain as a direct summand. In particular, we show in section I that the HFD, CHFD, and k—HFD properties are equivalent for rings of algebraic integers. In section II we extend the results of section I by constructing examples of Dedekind domains which are both CHFD and k—HFD but not HFD. In section In we explore the k-HFD property in detail when the class group of D is a torsion group. Finally in section IV we study the HFD, CHFD, and k—HFD properties in Dedekind domains with class group 11.