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Angiogenesis, the formation of a network of blood vessels, is a vital process in the growth of solid tumors as it delivers required nutrients and oxygen. Prior medical studies assert that angiogenesis is influenced by a number of parameters such as endothelial cell migration and proliferation, existence of tumor angiogenesis factors, oxygen, extracellular matrix components, etc. Along with the early developments and findings in the area of tumor angiogenesis, a field of research that has emerged uses mathematical models to interpret and predict the time-course of the crucial factors, as well as new capillary vessel formation, loop formation, and vessel branching. However, most of these early mathematical approaches rely on a small number of parameters; and the characteristics of blood flow, which are significant factors in tumor vessel formation, are neglected. Relatively new integrated models based on the impact of multiple crucial factors and blood flow have seen some success in elucidating the behavior of angiogenesis. Here we review the contributions, opportunities, progress, and challenges of mathematical and computational models for understanding of the tumor-induced angiogenesis, and also consider studies that apply mathematical models to represent blood flow and opportunities for the investigation of therapies and treatments. At the same time, we identify a need for the inclusion of endothelial cell shape and dynamics in models of tumor-induced angiogenesis. Particularly, cell-matrix, cell shape, and cell-cell interaction is necessary for the explanation of blood vessel formation.