This work presents a thorough study of the power-efficient task allocation problem in a heterogeneous multi-core processor for thermally constrained tasks with real-time constraints. A thermally constrained task is one which, if executed on a core at the maximum possible speed, results in the temperature of that core exceeding a given safe temperature limit. Our problem formulation is based on the well praised thermal model hotspot-4, leakage power and delay models with DVFS (Dynamic Voltage and Frequency Scaling). It accounts for the differences in power consumption of tasks and cores as well as the differences in the thermal characteristics of the functional blocks in the die. The model also considers various components of the package and the dependence of leakage power on the chip temperature. The solution to this problem is not so straight forward due to the fact that there is a nonlinear-circular dependency between the overall power dissipation and the temperature. It becomes further complicated with the consideration of real-time constraints of the task i.e. deadlines. The solution technique presented in this monograph consists of four interdependent steps.