Vaccination is one of the key strategies to stop the COVID-19 pandemic. This review aims to evaluate the current state of vaccine development and to determine the issues that merit additional research. We conducted a literature review of the development of COVID-19 vaccines, their effectiveness, and their use in special patient groups. To date, 140 vaccines are in clinical development. Vector, RNA, subunit, and inactivated vaccines, as well as DNA vaccines, have been approved for human use. Vector vaccines have been well studied prior to the COVID-19 pandemic; however, their long-term efficacy and approaches to scaling up their production remain questionable. The main challenge for RNA vaccines is to improve their stability during production, storage, and transportation. For inactivated vaccines, the key issue is to improve their immunogenicity and effectiveness. To date, it has been shown that the immunogenicity of COVID-19 vaccines directly correlates with their clinical efficacy. In view of the constant mutation, the emerging new SARS-CoV-2 variants have been shown to be able to partially escape post-vaccination immune response; however, most vaccines remain sufficiently effective regardless of the variant of the virus. One of the promising strategies to improve the effectiveness of vaccination, which is being studied, is the use of different platforms within a single vaccination course. Despite significant progress in the development and study of COVID-19 vaccines, there are many issues that require further research.