Shocks and cold fronts in galaxy clusters, the largest gravitationally bound systems in the universe, are astrophysical laboratories where we can study the microphysics of the intracluster medium (ICM), a very hot ($T\sim10^7$--$10^8$~K) plasma. Being the main baryon content of galaxy clusters, the ICM plays an important role in mediating the energy cascade from gravitational collapse during cosmological structure formation. It is also intricately linked to the evolution of the galaxies within. The scientific enquiries concerning the ICM range from fundamental physics questions to cosmological measurements.

In this dissertation, I demonstrate probing ICM microphysics by studying deep X-ray observations of two galaxy clusters, A520 and A2142. For A520, tests for thermal conduction, electron--ion equilibration timescale, and particle acceleration at the shock were carried out. For A2142, a test for the effective viscosity was performed using two apparent Kelvin-Helmholtz eddies along its southern cold front. Other interesting features were discovered and analyzed, such as a low gas fraction subcluster in the A520 outskirts, and X-ray deficient channels that could be plasma depletion sheets in both clusters.