In this paper, we study the almost sure-stability and δmoment stability of discrete-time jump linear systems with a finite state form process. Criteria for almost sure and δ-moment stability, which are necessary and sufficient conditions for (scalar) one-dimensional systems, are derived. The relationship between almost sure and 6-moment stability for (scalar) one-dimensional systems is also presented. If the system matrices commute, necessary and sufficient conditions for almost sure stability are obtained.