In dilute, magnetised plasmas exchanges of heat across magnetic field lines are strongly suppressed, as charged particles rapidly gyrate around magnetic field lines but only infrequently scatter due to Coulomb collisions. The anisotropic heat conduction, in the presence of a large-scale temperature gradient, destabilises otherwise stable entropy-stratified flows (according to the classic Schwarzschild criterion) via the magneto-thermal instability (MTI) and generates turbulence. The MTI is relevant in the outskirts of galaxy clusters, where it may supply a source of observed turbulence in the intra-cluster medium (ICM), and in the plasma of hot accretion flows around black holes and neutron stars, where the MTI can interact with the magneto-rotational instability (MRI) to produce complex dynamical cycles by attacking the temperature gradients produced by the MRI.

In the first part of the thesis, we take a fresh look at the problem of the MTI in the absence of shear and rotation, and construct a general theory that explains the MTI saturation mechanism and provides scalings and estimates for the turbulent levels. We simulate MTI turbulence with a Boussinesq code, SNOOPY, and find that in two dimensions the saturation mechanism involves an inverse cascade carrying kinetic energy from the short MTI injection scales to larger scales, where it is arrested by the stable entropy stratification; in three dimensions, on the other hand, most energy is dissipated at the same scale as its injection, and turbulent eddies are vertically elongated at or below the thermal conduction length, but relatively isotropic on larger scales. Similar to 2D, however, the saturated turbulent energy levels and the integral scale follow clear power-laws that depend on the thermal diffusivity, temperature gradient, and buoyancy frequency. We then show that our scaling laws are consistent with extant observations of ICM turbulence if the thermal conductivity is reduced by a factor of about ten from its Spitzer value.

In the second part of the thesis, we begin to address the question of how the MTI is modified in the presence of shear and rotation, and attempt to characterise the interplay between the MTI and the MRI in hot accretion flows. Our preliminary shearing-box studies show that for adiabatic, Keplerian flows the two instabilities manifest a hysteresis loop as the ratio of the shearing frequency to the MTI frequency is gradually increased. In a physical system where the large-scale background profiles are left free to evolve, this loop may be cycled dynamically and an MTI-dominated state alternate with an MRI-dominated state. For Keplerian but weakly (stably-)stratified flows, instead, we do not find evidence of hysteresis and for a critical value of shearing frequency the system transitions smoothly between the two regimes. Finally, we discuss the possible implications of these MTI-MRI cycles for X-ray binaries and their observed spectral variability.