Zero-difference balanced (ZDB) functions integrate a number of subjects in combinatorics and algebra, and have many applications in coding theory, cryptography and communications engineering. In this paper, three new families of ZDB functions are presented. The first construction, inspired by the recent work \cite{Cai13}, gives ZDB functions defined on the abelian groups $(\gf(q_1) \times \cdots \times \gf(q_k), +)$ with new and flexible parameters. The other two constructions are based on $2$-cyclotomic cosets and yield ZDB functions on $\Z_n$ with new parameters. The parameters of optimal constant composition codes, optimal and perfect difference systems of sets obtained from these new families of ZDB functions are also summarized.