In this paper, we rigorously analyze an HIV-1 infection model with CTL immune response and three time delays which represent the latent period, virus production period and immune response delay, respectively. We begin this model with proving the positivity and boundedness of the solution. For this model, the basic reproduction number $R_0$ and the immune reproduction number $R_1$ are identified. Moreover, we have shown that the model has three equilibria, namely the infection-free equilibrium $E_0,$ the infectious equilibrium without immune response $E_1$ and the infectious equilibrium with immune response $E_2.$ By applying fluctuation lemma and Lyapunov functionals, we have demonstrated that the global stability of $E_0$ and $E_1$ are only related to $R_0$ and $R_1.$ The local stability of the third equilibrium is obtained under four situations. Further, we give the conditions for the existence of Hopf bifurcation. Finally, some numerical simulations are carried out for illustrating the theoretical results.