The transport of solids is often encountered in well drilling, completion, well intervention and production processes. The process is affected by many variables and the complexity of the phenomena presents challenges to the field engineer who is trying to determine how these parameters affect solids-transport even if only one of these variables is changed during an operation. In the early stage, solids-transport studies mainly focused on finding the minimum critical velocity to prevent the formation of a stationary debris bed in the wellbore annulus for conventional rotary drilling with mud fluids. Over the past 10 years, the research and development focus has been on obtaining information related to the prediction of the equilibrium solids concentration profile along the whole well path during tripping in, on the wiper trip speed during tripping out, and on the prediction of the hole-cleaning time. Various correlations and models have been developed that would guide the field engineer to design and optimize the hole cleaning process. However, due to the complexity of the process, a comprehensive and proven model still does not exist at present.In this paper, based on 20 years of research experience, the authors present a critical review of the state of the art of previous approaches to studying the solids transport during both drilling and well interventions. It mainly focuses on theoretical studies. An overview of the experimental study of solids transport is given in a companion paper (Li et al. 2014). The methodology used to develop a model to simulate the hole cleaning process is also discussed. To date, completely satisfactory solids transport software, is still not available for the oil industry. Hopefully, this paper will provide a useful and helpful general guide for future research related to this subject and will aid in a better understanding of the lack of previous studies and what issues should be focused on in future research.