Abstract(#br)In this paper, it is shown that the problem of checking robust stability of linear parameter varying (LPV) systems is NP -hard, and therefore, it is rather unlikely to find polynomial-time solution procedures for this problem. In the frequency-domain structured uncertainty case, it is known that the robust stability problem is NP -hard (Toker and Özbay, 1995; Toker, 1995; Poljak and Rohn, 1993; Nemirovski, 1993; Braatz et al ., 1994), but allowing the uncertain blocks to be time varying gives a computationally tractable problem (Shamma, 1994; Poolla and Tikku, 1995), which can be solved by convex optimization techniques. In the parameteric uncertainty case, NP -hardness of the robust stability problem has been shown in (Poljak and Rohn, 1993; Nemirovski, 1993). The results of... this paper show that, allowing the uncertain parameters to be time varying, does not give a computationally simpler problem i.e. it remains NP -hard, and hence it is rather unlikely to find computationally tractable solution procedures for this problem. On the other hand, as far as the existence of an algorithm is considered, there is still no known (non-polynomial time) algorithm for the robust stability problem of linear parameter varying systems (Lagarias and Wang, 1995), and the well-known Tarski's theorem (Tarski, 1951) does not provide a solution procedure (Kozyakin, 1990). Recently, there has been some developments in the direction of constructing non-polynomial-time algorithms for a related problem, called the joint spectral radius (JSR) computation problem (Lagarias and Wang, 1995). We also comment on the use of these results for developing a non-polynomial-time algorithm for testing robust stability of linear parameter varying systems.