The low-energy nuclear response at finite-temperature significantly affects the radiative neutron capture reaction rates of the r-process nucleosynthesis. In order to address this topic, the first part of this study focuses on the response of compound nuclei or nuclei at finite temperature. The thermal nuclear response satisfies the Bethe-Salpeter equation (BSE) with the static and dynamical kernels of different origins. While the origin of the static kernel is the nearly instantaneous nucleon-meson interaction, the dynamical kernel is induced by the coupling between nucleons and phonons. The presence of singularities in the dynamical kernel makes the BSE unsolvable, however, a time projection technique known for the zero-temperature case allows for constructing a hierarchy of feasible approximations. In this study a temperature-dependent projection operator on the subspace of the imaginary time was found to generalize the method to finite temperatures. The method named the finite-temperature relativistic time blocking approximation (FT-RTBA), is implemented numerically to calculate the multipole responses of medium-mass and heavy nuclei. This study reveals common phenomena that occur for all thermal multipole responses: the disappearance of the high-frequency collective motion at very high temperature and arising prominent low-energy strength of thermal origin. The inclusion of pairing correlations and continuum effects is essential for an accurate microscopic description of the nuclear response of the exotic nuclei far from the valley of beta-stability and close to the drip-lines. Therefore, the second part of this study aims to extend the current zero-temperature nuclear response theory, which is based on the contact effective interactions between nucleons and takes into account the pairing correlations within the framework of the BCS approximation and exact coupling to the continuum. This extension involves the application of the time-blocking approximation in the coordinate space representation to incorporate the coupling between nucleons and phonons, which is the leading-order mechanism of the fragmentation of the nuclear multipole responses at both low- and high-frequency domains.