Ion cyclotron emission (ICE) was the first collective radiative instability, driven by confined fusion-born ions, observed from deuterium-tritium plasmas in JET and TFTR. ICE comprises strongly suprathermal emission, which has spectral peaks at multiple ion cyclotron harmonic frequencies as evaluated at the outer mid-plane edge of tokamak plasmas. The measured intensity of ICE spectral peaks scaled linearly with measured fusion reactivity in JET. In other large tokamak plasmas, ICE is currently used as an indicator of fast ions physics. The excitation mechanism for ICE is the magnetoacoustic cyclotron instability (MCI); in the case of JET and TFTR, the MCI is driven by a set of centrally born trapped fusion products, lying just inside the trapped-passing boundary in velocity space, whose drift orbits make large radial excursions to the outer mid-plane edge. Diagnostic exploitation of ICE in future experiments therefore rests in part on deep understanding of the MCI, and recent advances in computational plasma physics have led to substantial recent progress, reviewed here. Particle-in-cell simulations of the MCI, with fully kinetic ions and electrons, were reported in 2013, using plasma parameters for JET ICE observations. The hybrid approximation for plasma simulations, where ions are treated as particles and electrons as a neutralising massless fluid, was then applied and reported in 2014. These simulations extend previous studies deep into the nonlinear regime of the MCI, and corroborate predictions by linear analytical theory, thereby strengthening further the link to ICE measurements. ICE is a potential diagnostic for confined alpha-particles in ITER, where measurements of ICE could yield information on energetic ion behaviour supplementing that obtainable from other diagnostics. In addition, it may be possible to use ICE to study fast ion redistribution and loss due to MHD activity in ITER.