In this talk we consider several primeness properties of comodules and corings, transferred from primeness properties of modules and rings. Making use of the so called internal coproduct of fully invariant subcomodules, we extend the notion of (pre-)coprime coalgebras over base fields to pre-coprime (pre-cosemiprime) comodules for corings over arbitrary ground rings. We study also coendo-prime (coendo-semiprime) comodules, i. e. comodules that are prime (semiprime) as canonical modules over their rings of colinear endomorphisms, and prime (semiprime) comodules, i. e. modules that are prime (semiprime) as rational modules over the dual rings of their ground corings. Moreover we clarify the relations between these different primeness properties and simplicity (semisimplicity) and irreducibility of the comodules under consideration. The results we get are applied then to present, study and characterize corings satisfying these different primeness conditions and clarify their relations with simple (semisimple) and irreducible corings.