This paper deals with a special type of asymptotic (exponential) stability namely componentwise asymptotic (exponential) stability for 1-D and 2-D linear discrete-time systems. The main motivation for these results is the need. particularly felt in the evaluation in a more detailed manner of the dynamical behaviour of 1-D and 2-D linear discrete-time systems, Necessary and sufficient conditions for componentwise asymptotic (exponential) stability are given.