Absolute stability criteria for systems with multiple hysteresis non-linearities are given in this paper. It is shown that the stability guarantee is achieved with a simple two part test on the linear subsystem. If the linear subsystem satisfies a particular linear matrix inequality and a simple residue condition, then, as is proven, the non-linear system will be asymptotically stable. The main stability theorem is developed using a combination of passivity, Lyapunov and Popov stability theories to show that the state describing the linear system dynamics must converge to an equilibrium position of the non-linear closed loop system. The invariant sets that contain all such possible equilibrium points are described in detail for several common types of hystereses. The class of non-linearities covered by the analysis is very general and includes multiple slope-restricted memoryless non-linearities as a special case. Simple numerical examples are used to demonstrate the effectiveness of the new analysis in comparison to other recent results, and graphically illustrate state asymptotic stability.