This paper develops a family of Markov-switching GARCH (MSG) processes that not only encompasses some specifications in the literature, but also can be regarded as a Markov-switching version of the family of GARCH processes introduced by He and Teräsvirta (J. Econometrics, 1999, 92, 173–192). Some structural properties of this new family of MSG processes are considered. First, a sufficient and necessary condition for the existence of the strictly stationary solution of the family of MSG processes is presented. Moreover, we also give a sufficient and necessary condition for the existence of the strictly stationary solution of the family of MSG processes with finite δ-order moment. In particular, a definition of so-called family of integrated MSG processes is introduced and its stationarity is also discussed. Next, the general conditions for the existence of any order moment of the family of MSG processes are derived. Finally, by means of a new renewal theorem, we describe the tail of the marginal distribution of the family of MSG processes.