Viruses remain a global threat to animals, plants, and humans. The type 1 human immunodeficiency virus (HIV-1) is a member of the retrovirus family and carries an RNA genome, which is reverse transcribed into viral DNA and further integrated into the host-cell DNA for viral replication and proliferation. The RNA structures from the HIV-1 genome provide valuable insights into the mechanisms underlying the viral replication cycle. Moreover, these structures serve as models for designing novel therapeutic approaches. Here, we review structural data on RNA from the HIV-1 genome as well as computational studies based on these structural data. The review is organized according to the type of structured RNA element which contributes to different steps in the viral replication cycle. This is followed by an overview of the HIV-1 transactivation response element (TAR) RNA as a model system for understanding dynamics and interactions in the viral RNA systems. The review concludes with a description of computational studies, highlighting the impact of biomolecular simulations in elucidating the mechanistic details of various steps in the HIV-1's replication cycle.