Fusion physics focuses on the complex behaviour of hot plasmas confined by magnetic fields with the ultimate aim to develop a fusion power plant. In the future generation of tokamaks like ITER, the power generated by the fusion reactions substantially exceeds the external input power (Pfusion}/Pin ⩾ 10). When this occurs one speaks of a burning plasma. Twenty per cent of the generated fusion power in a burning plasma is carried by the charged alpha particles, which transfer their energy to the ambient plasma in collisions, a process called thermalization. A new phenomenon in burning plasmas is that the alpha particles, which form a minority but carry a large fraction of the plasma kinetic energy, can collectively drive certain types of magneto-hydrodynamic (MHD) modes, while they can suppress other MHD modes. Both types of MHD modes can have desirable effects on the plasma, as well as be detrimental to the plasma. For example, the so-called sawtooth instability, on the one hand, is largely responsible for the transport of the thermalized alpha particles out of the core, but, on the other hand, may result in the loss of the energetic alphas before they have fully thermalized. A further undesirable effect of the sawtooth instability is that it may trigger other MHD modes such as neoclassical tearing modes (NTMs). These NTMs, in turn, are detrimental to the plasma confinement and in some cases may even lead to disruptive termination of the plasma. At the edge of the plasma, finally, so-called edge localized modes or ELMs occur, which result in extremely high transient heat and particle loads on the plasma-facing components of a reactor. In order to balance the desired and detrimental effects of these modes, active feedback control is required. An additional complication occurs in a burning plasma as the external heating power, which is nowadays generally used for plasma control, is small compared to the heating power of the alpha particles. The scientific challenge in the field of burn control is to find the proper balance between desired and detrimental effects of the various MHD modes and to develop the methods and tools for active feedback control of MHD modes in burning plasmas. Therefore, it is necessary to understand the dynamics of the system, in this case the mutual interactions between the fast alpha particles and the MHD instabilities. Since burning plasmas do not yet exist, the relevant experimental work until ITER comes into full operation needs to be largely based on alpha-particle simulation experiments in which the alpha particles are accelerated to high energies by means of special heating techniques. The precise conditions of a burning plasma can be only partly mimicked in present tokamaks. Hence, also a detailed computational modelling effort is needed, in order to understand the impact of findings in present machines for those of the future.In 2011 two dedicated workshops were devoted to MHD control. Firstly, there was a workshop on Control of Burning Plasmas that took place from 21–25 March 2011 at the Lorentz Centre in Leiden, The Netherlands. Secondly, the 480th Wilhelm and Else Heraeus Seminar that took place from 16–18 June in Bad Honnef, Germany was devoted to Active Control of Instabilities in Hot Plasmas. This special issue presents a collection of papers that have been presented at the two workshops, along with a few papers that are the result of an open call to contribute to this special issue.