For 30 years two general mechanisms have competed to explain somatic hypermutation of immunoglobulin (Ig) genes. The first, the DNA-based model, is focused only on DNA substrates. The modern form is the Neuberger "DNA Deamination Model" based on activation-induced cytidine deaminase (AID) and short-patch error-prone DNA repair by DNA Polymerase-η operating around AID C-to-U lesions. The other is an RNA-based mechanism or the "Reverse Transcriptase Model" of SHM which produces strand-biased mutations at A:T and G:C base pairs. This involves error-prone cDNA synthesis via an RNA-dependent DNA polymerase copying the Ig pre-mRNA template and integrating the now error-filled cDNA copy back into the normal chromosomal site. The modern form of this mechanism depends on AID dC-to-dU lesions and long tract error-prone cDNA synthesis of the transcribed strand by DNA Polymerase-η acting as a reverse transcriptase. The evidence for and against each mechanism is critically evaluated. The conclusion is that all the SHM molecular data gathered since 1980 supports directly or indirectly the RNA/RT-based mechanism. All the data and critical analyses are systematically laid out so the reader can evaluate this conclusion for themselves. Recently we have investigated whether similar RNA/RT-based mutator mechanisms explain how de novo mutations arise in somatic tissues (cancer genomes). The data analyses indeed suggest that cancers arise via dysregulated "Ig-like SHM responses" involving rogue DNA and RNA deaminations coupled to genome-wide RT events. Further, Robyn Lindley has recently shown that the strand-biased mutations in cancer genome genes are also in "codon-context." This has been termed Targeted Somatic Mutation (TSM) to highlight that mutations are far more targeted than previously thought in somatic tissues associated with disease. The TSM process implies an "in-frame DNA reader" whereby DNA and RNA deaminases at transcribed regions are guided in their mutagenic action, by the codon reading frame of the DNA.