Summary: Human T-cell leukaemia virus type I (HTLV-I) preferentially infects the \(CD4^{+}\) T cells. The HTLV-I infection causes a strong HTLV-I specific immune response from \(CD8^{+}\) cytotoxic T cells (CTLs). The persistent cytotoxicity of the CTL is believed to contribute to the development of a progressive neurologic disease, HTLV-I associated myelopathy/tropical spastic paraparesis (HAM/TSP). We investigate the global dynamics of a mathematical model for the CTL response to HTLV-I infection in vivo. To account for a series of immunological events leading to the CTL response, we incorporate a time delay in the response term. Our mathematical analysis establishes that the global dynamics are determined by two threshold parameters \(R_{0}\) and \(R_{1}\), basic reproduction numbers for viral infection and for CTL response, respectively. If \(R_{0}\leq 1\), the infection-free equilibrium \(P_{0}\) is globally asymptotically stable, and the HTLV-I viruses are cleared. If \(R_{1}\leq 1<R_{0}\), the asymptomatic-carrier equilibrium \(P_{1}\) is globally asymptotically stable, and the HTLV-I infection becomes chronic but with no persistent CTL response. If \(R_{1}>1\), a unique HAM/TSP equilibrium \(P_{2}\) exists, at which the HTLV-I infection is chronic with a persistent CTL response. We show that the time delay can destabilize the HAM/TSP equilibrium, leading to Hopf bifurcations and stable periodic oscillations. Implications of our results to the pathogenesis of HTLV-I infection and HAM/TSP development are discussed.