More than 40 years after the first report of AIDS, and despite tremendous progress in treatment and prevention against HIV infection, no vaccine or cure is available. There is a strong need for better understanding host immune responses and viral reservoirs in tissues. Recent results in HIV cure research indicate that combination of strategies might be necessary to achieve durable remission or eradication of HIV. However, the number of combinations that can be tested are limited and need optimal guidance. Animal models allow deeper insights into tissue host-virus interactions and to perform mechanistic as well as proof-of-concept studies. The main NHP models correspond to macaques, where SIVmac infection displays similar replication profiles as HIV in humans, or macaques infected by a recombinant virus (SHIV) expressing HIV-1 ENV for analyzing the efficacy of bnAbs. Lessons for viral reservoir control and tissue damage protection can also be obtained from studying the natural host of SIV (African green monkeys, sooty mangabeys). Of note, no animal model is perfect. Examples of pitfalls will be discussed, as well as limitations of NHP models with regard to human studies. NHP models allowed nonetheless to significantly increase our understanding of HIV pathogenesis in fundamental aspects. In addition, studies with NHP models also gave important insights into mechanisms of viral reservoir establishment and control, contributing to HIV cure research. NHP models also sometimes revealed informative for the development of novel explorative approaches toward HIV cure. Recently, the role of innate immunity, in particular the role of NK cells, for viral reservoir control within tissues has gained increased interest. NK cells have been known for long for being able to eliminate abnormal and/or infected cells, through distinct mechanisms, including ADCC. Studies of macaques and AGM chronically infected by SIV have revealed the presence of adaptive and memory NK cells. The differentiation into potent NK cells might be tissue environment specific. Cytokine-induced, MHC-E dependent SIV-Env specific adapted and highly differentiated NK cells have been shown to be associated with reduction of replication-competent virus in lymph nodes from SIV-infected macaques under ART and with a delay of viral rebound after treatment interruption. Insights on tissue-specific immune responses and NK cells in NHP models with regard to their impact for HIV cure research will be discussed.