Many direct numerical simulations of spherical hot spot auto-ignitions, with different fuels, have identified different auto-ignitive regimes. These range from benign auto-ignition, with pressure waves of small amplitude, to super-knock with the generation of damaging over-pressures. Results of such simulations are generalised diagrammatically, by plotting boundary values of xi, the ratio of acoustic to auto-ignition velocity, against s. This latter parameter is the residence time of the developing acoustic wave in the hot spot of radius r(0), namely r(o)/a, normalised by the excitation time for the chemical heat release, tau(e). This ratio controls the energy transfer into the developing acoustic front. A third relevant parameter involves the product of the activation temperature, E/R, for the auto-ignition delay time, normalised by the mixture temperature. T, the ratio, tau(i)/tau(e), and the dimensionless hot spot temperature gradient, (partial derivative In T/partial derivative(r) over bar), where (r) over bar is a dimensionless radius. These parameters define the boundaries of regimes of thermal explosion, subsonic auto-ignition, developing detonations, and non-auto-ignitive deflagration, in plots of xi against epsilon. The regime of developing detonation forms a peninsula and contours, throughout the field. The product parameter (E/RT)(tau(i)/tau(e))/partial derivative In T//partial derivative(r) over bar expresses the influences of hot spot temperature gradient and fuel characteristics, and a unique value of it might serve as a boundary between auto-ignitive and deflagrative regimes. Other combustion regimes can also be identified, including a mixed regime of both auto-ignitive and "normal" deflagrative burning. The paper explores the performances of a number of different engines in the regimes of controlled auto-ignition, normal combustion, combustion with mild knock and, ultimately, super-knock. The possible origins of hot spots are discussed and it is shown that the dissipation of turbulent energy alone is unlikely to lead directly to sufficiently energetic hot pots. The knocking characterisation of fuels also is discussed. (C) 2016 The Combustion Institute. Published by Elsevier Inc. All rights reserved.