The study of modules by properties of their homomorphisms has long been of interest. Hiremath, introduced the notion of Hopfian modules. A bit later, Varadarajan, introduced the concept of co-Hopfian modules. These modules and several generalizations of them are extensively studied also by several authors. In this paper, we present new properties, characterizations and examples of these modules and of their generalizations. In the context of the co-Hopficity of modules, we also study the analogue of basis theorem, that is, the transfer of co-Hopficity to certain polynomial extensions.
