This invention generally relates to a novel composition of RNA/mRNA medicines as well as vaccines produced by using replicase- and/or RNA-dependent RNA polymerase (RdRp)-mediated RNA cycling reaction (RCR). The present invention is useful for developing a variety of self-amplifying RNA/mRNA (samRNA) medicines and vaccines containing at least a replicase/RdRp-binding site in the 5′- or 3′-end, or both, of any desired RNA molecule, including but not limited to antisense RNA (aRNA), small interferring RNA (siRNA), short hairpin RNA (shRNA), microRNA (miRNA)/miRNA precursor, long non-coding RNA (lnRNA) and mRNA. These RNA molecules can be either in single-stranded or in double-stranded, or mixed, conformation. The samRNA so obtained is useful not only for producing RNA-based vaccines and/or medicines but also for generating the mRNA-associated proteins, peptides, and/or antibodies under a proper in-vitro or in-cell translation condition. The replicase/RdRp-binding sites used in samRNA are derived or modified from coronaviral (e.g. COVID-19) and/or hepatitis C viral (HCV) RNA-dependent RNA polymerases (RdRp) in either single-stranded or double-stranded compositions.