Tensor abelian categories provide a framework for studying both the additive (abelian) and the multiplicative (monoidal) structure of categories like abelian grups, modules over rings, chain complexes, (dierential) graded modules, quasi-coherent sheaves and functor categories,even in the non-commutative setting. In the rst paper, we prove in this framework a classic theorem of Lazard and Govorov which states that at modules are precisely the direct limit closure of the nitely generated projective modules. The general result reproves this and other ad hoc examples and provide new results in other categories including the category of dierential graded modules. In the second paper we study quiver representations in such categories and characterize various classes of representations. This again generalizes characterizations in R-Mod, but provides new insight even in this case. In the last paper we study a generalization of the prime ideal spectrum in this setting, namely the atom spectrum. This has many good theoretical properties but concretecalculations are few. We provide a method for calculating this with several concrete examples.