Influenza virus, respiratory syncytial virus, human metapneumovirus, parainfluenza virus, coronaviruses, and rhinoviruses are among the most common viruses causing mild seasonal colds. These RNA viruses can also cause lower respiratory tract infections leading to bronchiolitis and pneumonia. Young children, the elderly, and patients with compromised cardiac, pulmonary, or immune systems are at greatest risk for serious disease associated with these RNA virus respiratory infections. In addition, swine and avian influenza viruses, together with severe acute respiratory syndrome-associated and Middle Eastern respiratory syndrome coronaviruses, represent significant pandemic threats to the general population. In this review, we describe the current medical need resulting from respiratory infections caused by RNA viruses, which justifies drug discovery efforts to identify new therapeutic agents. The RNA polymerase of respiratory viruses represents an attractive target for nucleoside and nucleotide analogs acting as inhibitors of RNA chain synthesis. Here, we present the molecular, biochemical, and structural fundamentals of the polymerase of the four major families of RNA respiratory viruses: Orthomyxoviridae, Pneumoviridae/Paramyxoviridae, Coronaviridae, and Picornaviridae. We summarize past and current efforts to develop nucleoside and nucleotide analogs as antiviral agents against respiratory virus infections. This includes molecules with very broad antiviral spectrum such as ribavirin and T-705 (favipiravir), and others targeting more specifically one or a few virus families. Recent advances in our understanding of the structure(s) and function(s) of respiratory virus polymerases will likely support the discovery and development of novel nucleoside analogs.