Hepatitis C virus (HCV) is a globally-dispersed agent of chronic hepatitis with a significant public health threat, affecting over 110 million individuals throughout the world. The increased risk for chronicity after exposure and the lack of a protective vaccine make HCV is a leading infectious cause of cirrhosis, liver failure requiring transplantation and hepatocellular carcinoma. The replicative process and infection dynamics in the host enable HCV to generate an array of closely-related but non-identical genetic variants known as quasispecies in the infected individuals. Pathogenesis and outcome in HCV infections are directly affected by the virus genetic heterogeneity, reflected as the emergence of quasispecies in infected individuals. The evolution of these highly-diverse viral populations in the host directly influences the disease course, via providing a pool of variants capable of resuming viral replication under extrinsic and/or intrinsic selective pressures. Viral quasispecies go through several alterations during the course of the infection, and provide a background for the selection of escape mutants from the host humoral and cell-mediated immune responses and antiviral treatment. Supported by the robust next generation sequencing techniques, recent studies have provided significant insights on the genomic diversity and progression as well as on the origin and the epidemiology of HCV. This review provides an overview of the mechanisms of HCV genetic variability, and the interactions with the host, that affects clinical disease, covering viral and host determinants of humoral and cell-mediated immune responses, alterations during the early and late stages of the infection and disease progression leading to chronicity. In addition, current findings in virus evolution and epidemiology were briefly interpreted from the interspecies and population perspectives. The impact of viral genomic heterogeneity on antiviral treatment in the era of direct-acting agents is also discussed, along with an overview of current methods employed for the characterization of viral diversity.