This chapter provides a critical survey of emergence definitions both from a conceptual and formal standpoint. The notions of downward / backward causation and weak / strong emergence are specially discussed, for application to complex social system with cognitive agents. Particular attention is devoted to the formal definitions introduced by (Müller 2004) and (Bonabeau & Dessalles, 1997), which are operative in multi-agent frameworks and make sense from both cognitive and social point of view. A diagrammatic 4-Quadrant approach, allow us to understanding of complex phenomena along both interior/exterior and individual/collective dimension.