Sheep and goats can efficiently convert low quality forage into high-quality meat which contains specific nutrients and quality traits. Carcass traits and quality attributes of sheep and goat meat depend upon several factors and one of most effective strategies amongst these is feeding regimens. In this review, the major aspects of feeding regimens affecting growth rate, carcass traits and quality attributes of sheep and goat meat are thoroughly discussed, with a particular focus on physical-chemical composition, flavor profile, and fatty acid (FA) profile. Grazing lambs and kids receiving concentrate or under stall-feeding systems had greater average daily gain and carcass yield compared with animals reared on pasture only. However, growth rate was higher in lambs/kids grazing on pastures of improved quality. Moreover, the meat of grazing lambs receiving concentrate had more intense flavor, intramuscular fat (IMF) content, and unhealthy FA composition, but comparable color, tenderness, juiciness, and protein content compared to that of lambs grazed on grass only. In contrast, meat of concentrate-fed lambs had more intense color, greater tenderness and juiciness, IMF and protein contents, and lower flavor linked to meat. Additionally, the meat of kids grazed on concentrate supplementation had higher color coordinates, tenderness, IMF content and unhealthy FA composition, whereas juiciness and flavor protein content were similar. In contrast, kids with concentrate supplementation had superior color coordinates, juiciness, IMF content and unhealthy FA composition, but lower tenderness and flavor intensity compared to pasture-grazed kids. Thus, indoor-finished or supplemented grazing sheep/goats had higher growth rate and carcass quality, higher IMF content and unhealthy FA composition compared to animals grazed on grass only. Finally, supplementation with concentrate increased flavor intensity in lamb meat, and improved color and tenderness in kid meat, whereas indoor-fed sheep/goats had improved color and juiciness as well as reduced flavor compared to pasture-grazed animals.