Nuclear density functional theory (DFT) is the method of choice to study the nuclear properties of medium-mass and heavy nuclei. This dissertation employs the Skryme Hartree-Fock-Bogoliubov (HFB) approach to study nuclear reflection-asymmetric deformations and collective rotation. Nuclear ground states with stable reflection-asymmetric shapes, predicted by theory, have been confirmed experimentally. To explore the microscopic origin of reflection-asymmetric nuclear shapes, we applied the density expansion method to decompose the total HFB energy into different multipolarities. We demonstrated that the reflection-asymmetric deformation is driven by the isoscalar part of the interaction energy. We also confirmed the importance of high-multipolarity fields for stabilizing reflection-asymmetric deformations. The nucleon localization function (NLF) has been successfully applied to characterize nuclear shell structure and collective motion. In our work, we extended the application of NLF to study the nuclear response to fast rotation. By solving the cranked harmonic-oscillator and comparing it with cranked Hartree-Fock results, we defined the simplified localization measure and demonstrated its usefulness as an indicator of nuclear rotation. The above nuclear DFT calculations were performed using existing HFB solvers. However, the current HFB solvers are deficient in the study of exotic nuclei whose properties are strongly affected by the quasiparticle continuum space. For this purpose, we developed a three-dimensional Skyrme-HFB solver HFBFFT in the coordinate-space representation using the canonical basis approach. We implemented the soft energy cutoff and pairing annealing to solve the problem of pairing collapse; a sub-iteration method to improve the convergence, and an algorithm to restore the Hermiticity of differential operators brought by Fourier-transform-based differentiation. The accuracy and performance of HFBFFT were tested by benchmarking it against other HFB codes, both spherical and deformed, for a set of well-bound and weakly-bound nuclei.
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