The prevalent common approach among sustainability analysis tools is to examine if some preconceived notions of what is acceptable, i.e., “sustainable”, hold true over a specific period of time. For example, within the commercial context, Life Cycle Analysis (LCA) inspects whether the environmental impacts associated with all stages of a commercial product or process are within an acceptable range. Expanding on this idea, this thesis aims to put forward a new mathematical viewpoint towards sustainability analysis of cases involving dynamical systems. Additionally, the idea of sustainability synthesis of such cases is presented for the first time. To this end, the Sustainability over Sets (SOS) and Sustainizability over Sets (SIZOS) concepts have been introduced as sustainable system analysis and synthesis tools, respectively. These concepts provide rigorous definitions of sustainability with a wide range of applications, and bring forward associated mathematical machinery enabling sustainability analysis of complex multidimensional systems and a methodology for sustainable system synthesis.By defining a system as an assemblage of interrelated components, SOS utilizes the concept of dynamical system theory and the mathematical idea of invariant sets to construct a conceptual framework within which the time evolution of a system can be analyzed and to provide an answer to its sustainability status independent of the examiner. This framework first formalizes the acceptable levels of sustainable behavior as sets in the system’s state space. It then translates the satisfaction of these sustainability criteria over an infinite time to the requirement that the system state trajectories initiated within the set remain within that set forever. To enable this analysis, necessary and sufficient mathematical criteria are first introduced for lumped systems described by autonomous ordinary differential equations (ODEs). Furthermore, these mathematical criteria are expanded to sustainability analysis of systems described by parabolic partial differential equations (PDEs) defined on a bounded and connected domain. Having established the methodology for sustainability analysis, Sustainizability Over Sets (SIZOS) refers to the existence of allowable external actions and/or design changes that can render sustainable over a defined set an unsustainable system. This novel concept provides necessary and sufficient criteria for sustainability synthesis of time variant forced systems. SOS and SIZOS concepts' wide range of applications is demonstrated through various lumped and spatially distributed cases with two to eleven dimensions. These systems exhibit complex behaviors such as oscillation or chaos. One such example is an isothermal biological CSTR, modeled through idealized cell and substrate balance equations and Monod kinetics. SOS analysis identified this system as unsustainable over rectangular sets regardless of its parameter values. Subsequently, this system has been made sustainable by manipulating the reactor's Damkohler number through the allowable change in feed flow rate and concentration. Next, sustainability of a chaotic, tri-trophic food chain, an oscillatory eleven-dimensional food network, and a fire-prone tree-grass system is assessed. Finally, by including the spatial heterogeneity within the tree-grass dynamics, its sustainability status under various wildfire frequencies, and tree grass growth rates has been analyzed. Ultimately, this body of work serves as the foundation for further developing SOS and SIZOS concepts as a whole and presents a guiding approach for their mathematical implementation through various examples.