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Galaxy clusters are the most massive structures in the universe and, in the hierarchical pattern of cosmological structure formation, the largest objects in the universe form last. Galaxy clusters are thus interesting objects for a number of reasons. Three examples relevant to this thesis are:1. Constraining the properties of dark energy: Due to the hierarchical nature of structure formation, the largest objects in the universe form last. The cluster mass function is thus sensitive to the entire expansion history of the universe and can be used to constrain the properties of dark energy. This constraint complements others derived from the CMB or from Type Ia supernovae and provides an important, independent confirmation of such methods. In particular, clusters provide detailed information about the equation of state parameter w because they sample a large redshift range z ∼ 0 − 1.2. Probing galaxy formation: Clusters contain the most massive galaxies in the uni- verse, and the most massive black holes; because clusters form so late, we can still witness the assembly of these objects in the nearby universe. Clusters thus provide a more detailed view of galaxy formation than is possible in studies of lower-mass ob- jects. An important example comes from x-ray studies of clusters, which unexpectedly found that star formation in massive galaxies in clusters is closely correlated with the properties of the hot, virialized gas in their halos. This correlation persists despite the enormous separation in temperature, in dynamical time-scales, and in length-scales between the virialized gas in the halo and the star-forming regions in the galaxy. This remains a challenge to interpret theoretically.3. Developing our knowledge of dilute plasmas: The masses and sizes of galaxy clusters imply that the plasma which permeates them is both very hot (∼ 108 K) and very dilute (∼ 10−2 cm−3). This plasma is collisional enough to be considered a fluid, but collisionless enough to develop significant anisotropies with respect to the local magnetic field. This interesting regime is one of the frontiers in theoretical studies of fluid dynamics. Unlike other astrophysical environments of similar collisionality (e. g. accretion disk coronae), galaxy clusters are optically thin and subtend large angles on the sky. Thus, they are easily observed in the x-ray (to constrain thermal processes) and in the radio (to constrain non-thermal processes) and provide a wonderful environment to develop our understanding of dilute plasmas.This thesis studies the dynamics of the hot gas in galaxy clusters, which touches on all three of the above topics.Chapter 2 shows that galaxy clusters are likely to be unstable to a new, vigorous form of convection. As a dynamical process which involves thermodynamic and magnetic properties of the gas, this convection bears directly on our understanding of the physics of dilute plas- mas. Furthermore, by moving metals and thermal energy through the cluster, convection may change the cooling rate of the gas and thus significantly impact the process of galaxy formation. Cluster convection also impacts the use of clusters as cosmological probes. Con- vection may drive turbulence in clusters with mean Mach numbers of order-unity. This changes the force balance in clusters, decreasing the thermal energy of a cluster of a given mass. Current methods for using clusters to constrain dark energy rely on observational probes of the thermal energy as a proxy for total mass. The accuracy of these methods depends on how vigorous cluster convection is.Chapter 3 studies thermal instability in galaxy clusters. I argue that clusters are all likely to be thermally unstable, but that this instability only grows to large amplitude in a subset of systems. Later studies have applied this result to galaxy formation in clusters and shown that one can reproduce some features of the well-known non-self-similarity at the high mass end of the galaxy luminosity function.Chapters 4 and 5 extends my work on convection (and, eventually, thermal instability) to consider the cosmological context of galaxy formation. This work aims to remove any arbitrary initial and boundary conditions from my simulations and is an important step toward a self-consistent model for the plasma physics in clusters.