Integrated Assessment Models (IAMs) have emerged as key tools for building and assessing long term climate mitigation scenarios. Due to their central role in the recent IPCC assessments, and international climate policy analyses more generally, and the high uncertainties related to future projections, IAMs have been critically assessed by scholars from different fields receiving various critiques ranging from adequacy of their methods to how their results are used and communicated. Although IAMs are conceptually diverse and evolved in very different directions, they tend to be criticized under the umbrella of "IAMs". Here we first briefly summarise the IAM landscape and how models differ from each other. We then proceed to discuss six prominent critiques emerging from the recent literature, reflect and respond to them in the light of IAM diversity and ongoing work and suggest ways forward. The six critiques relate to (1) representation of heterogeneous actors in the models, (2) modelling of technology diffusion and dynamics, (3) representation of capital markets, (4) energy-economy feedbacks, (5) policy scenarios, and (6) interpretation and use of model results.