Arctic amplification (AA) is a coupled atmosphere-sea ice-ocean process. This understanding has evolved from the early concept of AA, as a consequence of snow-ice line progressions, through more than a century of research that has clarified the relevant processes and driving mechanisms of AA. The predictions made by early modeling studies, namely the fall/winter maximum, bottom-heavy structure, the prominence of surface albedo feedback, and the importance of stable stratification have withstood the scrutiny of multi-decadal observations and more complex models. Yet, the uncertainty in Arctic climate projections is larger than in any other region of the planet, making the assessment of high-impact, near-term regional changes difficult or impossible. Reducing this large spread in Arctic climate projections requires a quantitative process understanding. This manuscript aims to build such an understanding by synthesizing current knowledge of AA and to produce a set of recommendations to guide future research. It briefly reviews the history of AA science, summarizes observed Arctic changes, discusses modeling approaches and feedback diagnostics, and assesses the current understanding of the most relevant feedbacks to AA. These sections culminate in a conceptual model of the fundamental physical mechanisms causing AA and a collection of recommendations to accelerate progress towards reduced uncertainty in Arctic climate projections. Our conceptual model highlights the need to account for local feedback and remote process interactions within the context of the annual cycle to constrain projected AA. We recommend raising the priority of Arctic climate sensitivity research, improving the accuracy of Arctic surface energy budget observations, rethinking climate feedback definitions, coordinating new model experiments and intercomparisons, and further investigating the role of episodic variability in AA.