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The Fourier heat conduction model is valid for most macroscopic problems.However, it fails when the wave nature of the heat propagation or time lagsbecome dominant and the memory or/and spatial non-local effects significant --in ultrafast heating (pulsed laser heating and melting), rapid solidificationof liquid metals, processes in glassy polymers near the glass transitiontemperature, in heat transfer at nanoscale, in heat transfer in a solid statelaser medium at the high pump density or under the ultra-short pulse duration,in granular and porous materials including polysilicon, at extremely highvalues of the heat flux, in heat transfer in biological tissues.  In common materials the relaxation time ranges from $10^{-8}$ to $10^{-14}$sec, however, it could be as high as 1 sec in the degenerate cores of agedstars and its reported values in granular and biological objects varies up to30 sec. The paper considers numerous non-Fourier heat conduction models thatincorporate time non-locality for materials with memory (hereditary materials,including fractional hereditary materials) or/and spatial non-locality, i.e.materials with non-homogeneous inner structure.

The Zoo of Non-Fourier Heat Conduction Models