Livestock and livestock products are often examined by discipline (environment, economy) or by organizational level (farm, sector, territory, country), making it difficult to simultaneously assess the diversity of services and impacts generated by livestock farming systems. Based on the scientific literature, we consider livestock farming systems at the interaction between the ecological, technical and social systems. Doing this, we can identify factors (biotic interactions, system management, collective organization) at the origin of their impacts and services. We then propose a simplified representation, the "barn", which visualizes how livestock systems (described by land use, animal density.) interact with their environment along five interfaces (inputs, environment and climate, markets, labor and employment, social and cultural factors). Two key characteristics of livestock farming systems, their integration into food systems (globalized vs. territorialized) and the extent to which they rely on exogenous vs. endogenous inputs, allow proposing a grid to analyze sustainability pathways for each type of territory; we use this grid to position the case studies analyzed in this special issue.