Abstract Attraction (or asymptoticily) is defined for a general space-time system consisting of a set endowed with a preorder (the “flow”), a family of monotone mappings (the “time scale”) and two collections of subsets, or quasifilters (a kind of generalized neighborhood systems). Thus, the notion of attraction is presented under the weakest possible conditions which allow us its adaptation to a wide variety of special situations. Within this framework we state necessary and sufficient conditions for asymptoticity in terms of Liapunov-type functions. INDEX TERMS: General SystemsattractionasymptoticilyLiapunov functions.