The domestic goat (Capra aegagrus hircus), a mammalian species with high genetic merit for production of milk and meat, can be a tremendously valuable tool for transgenic research. This research is focused on the production and multiplication of genetically engineered or genome-edited cloned specimens by applying somatic cell nuclear transfer (SCNT), which is a dynamically developing assisted reproductive technology (ART). The efficiency of generating the SCNT-derived embryos, conceptuses, and progeny in goats was found to be determined by a variety of factors controlling the biological, molecular, and epigenetic events. On the one hand, the pivotal objective of our paper was to demonstrate the progress and the state-of-the-art achievements related to the innovative and highly efficient solutions used for the creation of transgenic cloned does and bucks. On the other hand, this review seeks to highlight not only current goals and obstacles but also future challenges to be faced by the approaches applied to propagate genetically modified SCNT-derived goats for the purposes of pharmacology, biomedicine, nutritional biotechnology, the agri-food industry, and modern livestock breeding.