Messenger RNA (mRNA) vaccines have recently emerged as a new type of vaccine technology, showing strong potential to combat the COVID-19 pandemic. In addition to SARS-CoV-2 which caused the pandemic, mRNA vaccines have been developed and tested to prevent infectious diseases caused by other viruses such as Zika virus, the dengue virus, the respiratory syncytial virus, influenza H7N9 and Flavivirus. Interestingly, mRNA vaccines may also be useful for preventing non-infectious diseases such as diabetes and cancer. This review summarises the current progresses of mRNA vaccines designed for a range of diseases including COVID-19. As epitope study is a primary component in the in silico design of mRNA vaccines, we also survey on advanced bioinformatics and machine learning algorithms which have been used for epitope prediction, and review on user-friendly software tools available for this purpose. Finally, we discuss some of the unanswered concerns about mRNA vaccines, such as unknown long-term side effects, and present with our perspectives on future developments in this exciting area.