This work is devoted to the stability of random-switching systems of differential equations. After presenting the formulation of random-switching systems, the notion of stability is recalled, and sufficient conditions in terms of the Liapunov function are presented. Then easily verifiable conditions for stability and instability of systems arising in approximation are established. Using a logarithm transformation, necessary and sufficient conditions are derived for systems that are linear in the continuous state component. Several examples are provided as demonstrations. Among other things, a somewhat different behavior from the well-known Hartman-Grobman theorem is observed.