In the context of present global changes, interest in understanding how systems respond to anthropogenic environmental pressures and stress has increased. Indices that characterize ecosystem state are helpful tools for the interpretation of ecosystem responses. The central question is how to link these responses to ecosystem structure and functioning and to quantify ecosystem persistence, resistance or resilience. Quantification and characterization of trophic networks by ecological network analysis (ENA) indices is proceeding rapidly, especially in the field of coastal ecology. In this contribution, we review several theories that relate ecosystem structure and function to stability. The structure and functioning of ecosystems change during the maturation of ecosystems. In the first section, the maturation of ecosystems is described using thermodynamics. In the second and third parts of this paper, we define some concepts for analysing structure and functioning of food webs and discuss their relation to stability. In the last section, we describe three ENA indices and their link to stability. We demonstrate that ENA provides powerful tools for describing local stability, combining quantitative and qualitative concepts. However, it remains incomplete for describing real conservation cases that combine local and global stability.