This paper consists of two main parts. In the first part, we provide a framework for stabilization of arbitrary (not necessarily compact) closed sets for sampled-data nonlinear differential inclusions via their approximate discrete-time models. We generalize [D. Nesic, A. R. Teel, and P. V. Kokotovic, Systems Control Lett., 38 (1999), pp. 259-270, Theorem 1] in several different directions: we consider stabilization of arbitrary closed sets, plants described as sampled-data differential inclusions, and arbitrary dynamic controllers in the form of difference inclusions. Our result does not require the knowledge of a Lyapunov function for the approximate model, which is a standing assumption in [D. Nesic and A. R. Teel, IEEE Trans. Automat. Control, 49 (2004), pp. 1103-1122] and [D. Nesic, A. R. Teel, and P. V. Kokotovic, Systems Control Lett., 38 (1999), pp. 259-270, Theorem 2]. We present checkable conditions that one can use to conclude semiglobal asymptotic stability, or global exponential stability (GES), of the sampled-data system via appropriate properties of its approximate discrete-time model. In the second part, we present sufficient conditions for stability of parameterized difference inclusions that involve various summability criteria on trajectories of the system to conclude global asymptotic stability (GAS) or GES, and they represent discrete-time counterparts of results given in [A. R. Teel, E. Panteley, and A. Loria, Math. Control Signals Systems, 15 (2002), pp. 177-201]. These summability criteria are not Lyapunov based, and they are tailored to be used within our above-mentioned framework for stabilization of sampled-data differential inclusions via their approximate discrete-time models. We believe that these tools will be a useful addition to the toolbox for controller design for sampled-data nonlinear systems via their approximate discrete-time models.