Antiretroviral therapy effectively suppresses, but does not eradicate HIV-1 infection. Persistent low-level HIV-1 can still be detected in plasma and cellular reservoirs even after years of effective therapy, and cessation of current treatments invariably results in resumption of viral replication. Efforts to eradicate persistent HIV-1 require a comprehensive examination of the quantity and genetic composition of HIV-1 within the plasma and infected cells located in the peripheral blood and tissues throughout the body. Single-molecule techniques, such as the single-copy assay and single-genome/proviral sequencing assays, have been employed to further our understanding of the source and viral dynamics of persistent HIV-1 during long-term effective therapy. The application of the single-copy assay, which quantifies plasma HIV-1 RNA down to a single copy, has revealed that viremia persists in the plasma and CSF after years of effective therapy. This low-level HIV-1 RNA also persists in the plasma following treatment intensification, treatment with latency reversing agents, cancer-related therapy, and bone marrow transplantation. Single-genome/proviral sequencing assays genetically characterise HIV-1 populations after passing through different selective pressures related to cell type, tissue type, compartment, or therapy. The application of these assays has revealed that the intracellular HIV-1 reservoir is stable and mainly located in CD4+ memory T cells. Moreover, this intracellular HIV-1 reservoir is primarily maintained by cellular proliferation due to homeostasis and antigenic stimulation, although cryptic replication may take place in anatomic sites where treatment is sub-optimal. The employment of single-genome/proviral sequencing showed that latency reversing agents broadly activate quiescent proviruses but do not clear the intracellular reservoir. Recently, full-length individual proviral sequencing assays have been developed and the application of these assays has revealed that the majority of intracellular HIV-1 DNA is genetically defective. In addition, the employment of these assays has shown that genetically intact proviruses are unequally distributed in memory T cell subsets during antiretroviral therapy. The application of single-molecule assays has enhanced the understanding of the source and dynamics of persistent HIV-1 in the plasma and cells of HIV-infected individuals. Future studies of the persistent HIV-1 reservoir and new treatment strategies to eradicate persistent virus will benefit from the utilization of these assays.