Recent progress in computational materials science in the area of semiconductor epitaxial growth is reviewed. Reliable prediction can now be made for a wide range of problems, such as surface reconstructions, adsorption-desorption behavior, and growth processes at realistic growth conditions, using our ab initio-based chemical potential approach incorporating temperature and beam equivalent pressure. Applications are examined by investigating the novel behavior during the hetero-epitaxial growth of InAs on GaAs including strain relaxation and resultant growth mode depending growth orientations such as (111)A and (001). Moreover, nanowire formation is also exemplified for adsorption-desorption behaviors of InP nanowire facets during selective-area growth. An overview of these issues is provided and the latest achievement are presented to illustrate the capability of the theoretical-computational approach by comparing experimental results. These successful applications lead to future prospects for the computational materials design in the fabrication of epitaxially grown semiconductor materials.