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In this work, the stability of nonautonomous linear dynamic systems on time scales is investigated and analyzed. A unified and extended version of Lyapunov’s direct method is developed and yields criteria for uniform stability and uniform expo-nential stability of a linear dynamic system. We investigate “slowly varying ” nonau-tonomous systems and provide a spectral condition on the system matrix sufficient for exponential stability. Perturbations of the unforced system are studied and an in-stability criterion is introduced. We develop a comprehensive, unified Floquet theory including Lyapunov transformations and their various stability preserving properties, as well as a unified Floquet theorem which establishes a canonical Floquet decom-position on time scales in terms of the generalized exponential function. We then use these results to study homogenous as well as nonhomogeneous periodic problems. Furthermore, we explore the connection between Floquet multipliers and Floquet ex-ponents via monodromy operators and establish a spectral mapping theorem on time scales. We conclude with several nontrivial examples to show the utility of this theory