In this paper we discuss stability and stabilization of continuous and discrete multidimensional input/output (IO) behaviors (of dimension r) which are described by linear systems of complex partial differential (resp., difference) equations with constant coefficients, where the signals are taken from various function spaces, in particular from those of polynomial-exponential functions. Stability is defined with respect to a disjoint decomposition of the r-dimensional complex space into a stable and an unstable region, with the standard stable region in the one-dimensional continuous case being the set of complex numbers with negative real part. A rational function is called stable if it has no poles in the unstable region. An IO behavior is called stable if the characteristic variety of its autonomous part has no points in the unstable region. This is equivalent to the stability of its transfer matrix and an additional condition. The system is called stabilizable if there is a compensator IO system such that the output feedback system is well-posed and stable. We characterize stability and stabilizability and construct all stabilizing compensators of a stabilizable IO system (parametrization). The theorems and proofs are new but essentially inspired and influenced by and related to the stabilization theorems concerning multidimensional IO maps as developed, for instance, by Bose, Guiver, Shankar, Sule, Xu, Lin, Ying, Zerz, and Quadrat and, of course, the seminal papers of Vidyasagar, Youla, and others in the one-dimensional case. In contrast to the existing literature, the theorems and proofs of this paper do not need or employ the so-called fractional representation approach, i.e., various matrix fraction descriptions of the transfer matrix, thus avoiding the often lengthy matrix computations and seeming to be of interest even for one-dimensional systems (at least to the author). An important mathematical tool, new in systems theory, is Gabriel's localization theory which, only in the case of ideal-convex (Shankar, Sule) unstable regions, coincides with the usual one. Algorithmic tests for stability, stabilizability, and ideal-convexity, and the algorithmic construction of stabilizing compensators, are addressed but still encounter many difficulties; see in particular the open problems listed by Xu et al.