By the early 1960s, evidence had accumulated that proteins were synthesized from special RNA copies of genes, named "messenger RNAs" (mRNAs), not directly from the stable RNAs found in the ribosomes of the cytoplasm. Yet, precisely how the protein chains were assembled along the RNA and, in particular, the relationship between the mRNAs and the ribosomes during protein synthesis, was obscure. In this account, I discuss how my laboratory found that multiple ribosomes traverse each mRNA, yielding the structures known as polysomes. This work led on to the first physical determination of the coding ratio, new insights into how protein chains are initiated, and an early suggestion that chloroplasts and mitochondria in eukaryotic cells might ultimately have been derived from symbiotic bacteria.