Tissue growth and regeneration are fundamental processes underpinning crucial physiological and pathological conditions: ranging from normal blood vessel network development, response to stem cells therapy and cancers. Modelling of such biological phenomena has been addressed through mathematical and algorithmic approaches. The former implements continuous representations based on differential equations. The latter exploit operational descriptions in the form of computing programs to represent and execute the models. Within this area, models that define the cell as the fundamental unit of model development, as well as discrete representations of different model entities, are important to plan in vitro experiments and to generate new testable hypotheses. This article reviews the application of algorithmic discrete models, with a focus on tissue growth and regeneration phenomena in the context of health and disease. The review begins with an overview of basic concepts, problems and approaches of computational discrete models. This will include a discussion of basic assumptions and design principles. An overview of key cell-driven approaches and examples of applications in tissue growth and regeneration is provided. The specification, implementation and analysis of a model are illustrated with a hypothetical example, which mimics the branching and sprouting patterns observed in blood vessel network development. The article concludes with a discussion of current challenges and recommendations.