Bovine colostrum (BC), the first milk produced from cows after parturition, is increasingly used as a nutritional supplement to promote gut function and health in other species, including humans. The high levels of whey and casein proteins, immunoglobulins (Igs), and other milk bioactives in BC are adapted to meet the needs of newborn calves. However, BC supplementation may improve health outcomes across other species, especially when immune and gut functions are immature in early life. We provide a review of BC composition and its effects in infants and children in health and selected diseases (diarrhea, infection, growth-failure, preterm birth, necrotizing enterocolitis (NEC), short-bowel syndrome, and mucositis). Human trials and animal studies (mainly in piglets) are reviewed to assess the scientific evidence of whether BC is a safe and effective antimicrobial and immunomodulatory nutritional supplement that reduces clinical complications related to preterm birth, infections, and gut disorders. Studies in infants and animals suggest that BC should be supplemented at an optimal age, time, and level to be both safe and effective. Exclusive BC feeding is not recommended for infants because of nutritional imbalances relative to human milk. On the other hand, adverse effects, including allergies and intolerance, appear unlikely when BC is provided as a supplement within normal nutrition guidelines for infants and children. Larger clinical trials in infant populations are needed to provide more evidence of health benefits when patients are supplemented with BC in addition to human milk or formula. Igs and other bioactive factors in BC may work in synergy, making it critical to preserve bioactivity with gentle processing and pasteurization methods. BC has the potential to become a safe and effective nutritional supplement for several pediatric subpopulations.