In this paper we introduce a special notion of Invariance Set for certain classes of dynamical systems: the concept has been inspired by our experience with models drawn from Biology. We claim that Box Invariance, that is, the existence of "boxed" invariant regions, is a characteristic of many biologically-inspired dynamical models, especially those derived from stoichiometric reactions. Moreover, box invariance is quite useful for the verification of safety properties of such systems. This paper presents effective characterization of this notion for linear and affine systems, the study of the dynamical properties it subsumes, computational aspects of checking for box invariance, and a comparison with related concepts in the literature. The concept is illustrated using two models from biology.