This thesis analyzes the evolutionary trajectories that drive the evolution of several RNA viruses. These viruses have been identified to be the leading causes of viral outbreaks and deaths in humans. Studying the mechanisms influencing their evolution could therefore produce vital information for controlling the spread of these viruses or for their eradication. The availability of huge sequence repositories and advancement in computing and sequencing technologies allows for the development of novel methods for understanding the evolution of viruses even during an on-going outbreak, epidemic or pandemic. In this study, I developed a method that incorporates phylogenetic and structural based techniques to study the evolution of drug resistance in (A) HIV-1 Pol proteins, (B) the evolutionary dynamics of the 2013 - 2016 EBOV outbreak and (C) the evolution of the A(H1N1) influenza virus amongst human, avian and swine species. Findings from this thesis show that though HIV-1 evolves differently in the presence and absence of drug selection pressure, the virus is generally constrained by the need to maintain viral protein structure stability. The virus achieves this by accumulating enabling mutations early in its evolutionary history in order to accommodate the emergence of drug resistance associated mutations, which are mostly destabilizing. I also show that although the 2013 - 2016 EBOV was evolving rapidly, early data indicated that it was not changing at the functional level and not adapting to the human host. This is because most of the mutations occur in either inter genic or intrinsically disordered regions, which are less constrained, while the structured bits are characterized by neutral impact mutations. This again suggests that the virus needs to maintain a stable protein structure in order to remain functional. I show that EBOV is relatively stably evolving and the major force driving its evolution is more of an epidemiologic rather than a molecular factor whereas HIV-1 is evolving adaptively and its evolution is driven by molecular processes. However, one residue change, A82V seems to have altered the ability of the virus to bind its human receptor. This suggests that adaptive or functional mutations (which are mostly destabilizing in nature) work hand in hand with enabling mutations in such a way that a virus can acquire a mutation that confers drug resistance or leads to a gain of function without compromising its fitness while also retaining its functions such as infectivity and transmissibility. This indicates that the mechanisms described above may be a general way through which viruses evolve. The methods developed in the study can easily be applied to studying the evolution of other viruses and other systems e.g. microorganisms and cancer cells. Even if selection analysis does not show positive selection or any mutations in functional site, my thesis has demonstrated that structural analysis will be very useful for identifying and also predicting mutations that could facilitate adaptation of viruses. Also the influenza study shows that though the A(H1N1) is evolving somewhat differently in the humans, avian and swine species, one thing they seem to have in common is that stability constrains their evolution. I also show that my findings based on the human A(H1N1) influenza virus is consistent with the other human viruses (HIV and EBOV) analyzed in this project work.