The drive for higher performance has led to greater integration and higher clock frequency of microprocessor chips. This translates into higher heat dissipation and, therefore, effective cooling of electronic chips is becoming increasingly important for their reliable performance. We systematically explore the limits for heat removal from a model chip in various configurations. First, the heat removal from a bare chip by pure heat conduction and convection is studied to establish the theoretical limit of heat removal from a bare die bound by an infinite medium. This is followed by an analysis of heat removal from a packaged chip by evaluating the thermal resistance due to individual packaging elements. The analysis results allow us to identify the bottlenecks in the thermal performance of current generation packages, and to motivate lowering of thermal resistance through the board-side for efficient heat removal to meet ever increasing reliability and performance requirements.