An intrinsic trajectory level approach without any recourse to an algebraic structure of a representation is utilized to develop a behavioral approach to robust stability. In particular it is shown how the controllable behavior can be constructed at the trajectory level via Zorn's lemma, and this is utilized to study the controllable-autonomous decomposition. Stability concepts are defined, and the relation between this framework and the well-known difficulties of classical input-output approaches to systems over the doubly infinite time axis are discussed. The gap distance is generalized to the behavioral setting via a trajectory level definition; and a basic robust stability theorem is established for linear shift invariant behaviors. The robust stability theorem is shown to provide an explicit robustness interpretation to the behavioral H(infinity) synthesis of Willems and Trentelmann.