Influenza A virus (IAV) is the sole cause of the unpredictable influenza pandemics and deadly zoonotic outbreaks and constitutes at least half of the cause of regular annual influenza epidemics in humans. Two classes of anti-IAV drugs, adamantanes and neuraminidase (NA) inhibitors (NAIs) targeting the viral components M2 ion channel and NA, respectively, have been approved to treat IAV infections. However, IAV rapidly acquired resistance against both classes of drugs by mutating these viral components. The adamantane-resistant IAV has established itself in nature, and a majority of the IAV subtypes, especially the most common H1N1 and H3N2, circulating globally are resistant to adamantanes. Consequently, adamantanes have become practically obsolete as anti-IAV drugs. Similarly, up to 100% of the globally circulating IAV H1N1 subtypes were resistant to oseltamivir, the most commonly used NAI, until 2009. However, the 2009 pandemic IAV H1N1 subtype, which was sensitive to NAIs and has now become one of the dominant seasonal influenza virus strains, has replaced the pre-2009 oseltamivir-resistant H1N1 variants. This review traces the epidemiology of both adamantane- and NAI-resistant IAV subtypes since the approval of these drugs and highlights the susceptibility status of currently circulating IAV subtypes to NAIs. Further, it provides an overview of currently and soon to be available control measures to manage current and emerging drug-resistant IAV. Finally, this review outlines the research directions that should be undertaken to manage the circulation of IAV in intermediate hosts and develop effective and alternative anti-IAV therapies.