Across the globe, the COVID-19 pandemic has impacted the lives of people. The mental health of the students of higher educational institutions (HEIs) was worst hit by the coronavirus’ multiple waves and further induced by the great lockdown. Among the students, the impact of COVID-19 depends on various vulnerability factors such as current age, educational status, low economic status, pre-existing mental health problems, and fear of infection. The systematic search was conducted on PubMed, PsycINFO, Scopus, Google Scholar, ResearchGate, and other databases from January 2020 to September 2022. A total of 80 articles were included in the review based on the eligibility criteria. High prevalence rates of anxiety (3.82–87.7%), depression (21.2–82.4%), and stress (11–81.2%) were reported by the HEI students during the COVID-19 outbreak in the United States of America, Ethiopia, Egypt, Greece, Kosovo, Jordan, China, Malaysia, Bangladesh, Indonesia, Pakistan, and India. A variety of risk factors were also associated with psychological distress such as female gender, young age group of 18–19 years, low socio-economic status, academic delays, history of medical/psychiatric illness, fear of infection, and excessive exposure to COVID-19-related news via social media. The psychological impact of COVID-19 on the mental health of HEI students is higher in magnitude. The review highlights the mental health issues of HEI students and also presents interventions and strategies to help HEI students at various levels.