Influenza is a highly contagious respiratory pathogen that continues to evolve and threaten both veterinary and human public health. Influenza A viruses are continually undergoing molecular changes through mutations, reassortment, and, in rare instances, recombination. While they generally cause benign enteric infection in their natural reservoir of wild aquatic birds, they can cause catastrophic and potentially lethal disease outbreaks in humans, domestic poultry, and pigs when they cross the host species barrier. The continuing circulation of highly pathogenic (HP) H5N1 influenza viruses in domestic poultry in parts of Eurasia and the emergence and global spread of pandemic H1N1 2009 are current examples of influenza evolution. The spread of both HP H5N1 and pandemic H1N1 to multiple hosts emphasizes the potential for continued evolution. In this review, we discuss the current understanding of influenza A virus structure and strategies of variation, with a specific focus on the HP H5N1 and pandemic H1N1 influenza viruses. Additionally, we attempt to identify the gaps in our knowledge of H5N1 and pandemic H1N1 influenza viruses. These gaps include (i) an understanding of the molecular determinants of influenza virus and the host that permit efficient transmissibility and pandemic potential, (ii) the urgent need for prospective surveillance in apparently healthy swine, (iii) the molecular determinants of high pathogenicity in poultry, pigs, and people, (iv) the genetic basis of host susceptibility, (v) antigenic variability, (vi) the use of vaccine to control influenza, (vii) the role of wild birds as the reservoir of highly pathogenic avian influenza, (viii) the problems with vaccines, (ix) seasonality, (x) co-infections, and (xi) anti-influenza drug resistance. Our failure to eradicate HP H5N1 globally and to explain why H5N1 does not transmit efficiently in humans while an H1N1 pandemic virus of swine origin spread globally in months are key examples that emphasize the critical need to bridge these knowledge gaps. Future directions in influenza research that will help us resolve each of the above-mentioned knowledge gaps include complete genomic and proteomic analysis of both the virus and the host with the prospect of designing new control strategies and the development of genetically resistant hosts.