Clusters of galaxies are the largest gravitationally-bound objects in the universe, and as such are useful as probes of both cosmology and astrophysics. To understand these objects as well as related problems in galaxy formation and black hole growth, we must understand the hot, dilute intracluster medium (ICM) that dominates the baryonic mass. In the cores of cool-core galaxy clusters, I will explore the balance between heating and cooling processes and explain when and how thermal instability occurs. The resulting cool gas can form filaments or feed a central black hole. Our results are in generally good agreement with observations. I will describe a degree of thermal self-regulation of clusters and hot halos that can be achieved from this process and its consequences. Finally, I will move to the cluster outskirts and highlight the role of conduction-driven convection in the outskirts of galaxy clusters. The non-thermal pressure support from this vigorous convection has implications for our interpretation of ongoing SZ surveys.