Achieving the Sustainable Development Goals (SDGs) is contingent on managing complex interactions that create synergies and trade-offs between different goals. Given the importance of interactions, it is necessary to understand the system mechanisms underpinning them to provide insight into their non-linear behaviours such as side-effects, delay, and acceleration. Prominent methods of SDG analysis that focus on sector-specific modelling or data-driven statistical correlation are insufficient for presenting an integrated view of interactions among many goals. These methods are also often too technically complex or heavily data-driven to provide decision-makers with a simple practical tool and easily actionable and understandable results. To address this gap, we introduce a systems approach for analysing the SDGs that generalises a number of recurring interactions with unique structures and behaviours termed archetypes. We present eight interaction archetypes as thinking aids to conceptualise and analyse some of the important synergies and trade-offs, supported by several empirical studies related to the SDGs (e.g., poverty, food, well-being, water, energy, housing, climate, land-use). We also discuss how this approach can be operationalised in practice and what opportunities and challenges are ahead. Interaction archetype analysis advances sustainability science by giving researchers and policy-makers a diagnostic tool to identify fundamental mechanisms of barriers or policy resistance to SDG achievement, a comparative tool that can enhance knowledge transfer about the SDGs between different cases which share similar causal characteristics in a more coordinated way, and a prospective tool to design synergistic and transformational solutions for sustainable development.