Modules in which every proper submodule(proper nonzero submodule) is prime are called fully prime(almost fully prime) and with some other related notions are fully investigated. It is shown that over a commutative ring R, an R−module M is fully prime(fully semiprime) if and only if M is a homogeneous semisimple(co-semisimple) module. This in particular shows that a f.g. R−module M is co-semisimple if and only if R/Ann(M) is a regular(von-Neumann) ring. Modules in which nonzero direct summands are prime are also characterized. When R is a one-dimensional Noetherian domain we determine all modules in which the zero submodule is the only prime(semiprime) submodule. Finally, we observe that R is a Max-ring if and only if every R−module contains a prime (semiprime) submodule.