Abstract: The establishment of a concrete-damage-constitutive model is of great importance to characterize the behavior of concrete and understand the essence of concrete. The research history and the latest progress of concrete damage constitutive models in worldwide were summarized from deterministic and random aspects. The different definition methods, the evolution laws, and the physical mechanism of damage in the literatures are discussed. In addition, by comparing main characteristics such as nonlinearity, stochasticity and strain rate related, some critical issues need to be resolved for the accurate prediction of the concrete behavior are proposed, which provides a reference for establishing a more-reasonable and convenient concrete-damage-constitutive model.