Semiconductor scaling technology has led to a sharp growth in transistor counts. This has resulted in an exponential increase on both power dissipation and heat flux (or power density) in modern microprocessors. These microprocessors are integrated as the major components in many modern embedded devices, which offer richer features and attain higher performance than ever before. Therefore, power and thermal management have become the significant design considerations for modern embedded devices. Dynamic voltage/frequency scaling (DVFS) and dynamic power management (DPM) are two well-known hardware capabilities offered by modern embedded processors. However, the power or thermal aware performance optimization is not fully explored for the mainstream embedded processors with discrete DVFS and DPM capabilities. Many key problems have not been answered yet. What is the maximum performance that an embedded processor can achieve under power or thermal constraint for a periodic application? Does there exist an efficient algorithm for the power or thermal management problems with guaranteed quality bound? These questions are hard to be answered because the discrete settings of DVFS and DPM enhance the complexity of many power and thermal management problems, which are generally NP-hard. The dissertation presents a comprehensive study on these NP-hard power and thermal management problems for embedded processors with discrete DVFS and DPM capabilities. In the domain of power management, the dissertation addresses the power minimization problem for real-time schedules, the energy-constrained make-span minimization problem on homogeneous and heterogeneous chip multiprocessors (CMP) architectures, and the battery aware energy management problem with nonlinear battery discharging model. In the domain of thermal management, the work addresses several thermal-constrained performance maximization problems for periodic embedded applications. All the addressed problems are proved to be NP-hard or strongly NP-hard in the study. Then the work focuses on the design of the off-line optimal or polynomial time approximation algorithms as solutions in the problem design space. Several addressed NP-hard problems are tackled by dynamic programming with optimal solutions and pseudo-polynomial run time complexity. Because the optimal algorithms are not efficient in worst case, the fully polynomial time approximation algorithms are provided as more efficient solutions. Some efficient heuristic algorithms are also presented as solutions to several addressed problems. The comprehensive study answers the key questions in order to fully explore the power and thermal management potentials on embedded processors with discrete DVFS and DPM capabilities. The provided solutions enable the theoretical analysis of the maximum performance for periodic embedded applications under power or thermal constraints.