The possibility of shocks was first suggested by Stokes in 1848, but no consistent theory emerged for another almost 30 years. Indeed, severe criticism by Lord Kelvin and Lord Rayleigh led him to finally discard this idea. In the meantime, algebraic shock conditions had been derived by Rankine and Hugoniot, and their names have come to be associated with the shock conditions generally. As Thompson (Compressible-fluid dynamics, McGraw-Hill, New York, 1972) concludes, “Stokess claim to recognition for his discovery has been diverted by circumstance.” Starting with a brief discussion of these early developments, the present review paper will then concentrate on the important question of how to select from all formally possible solutions of the RankineHugoniot jump relations those which are physically realizable solutions. This is at the core of the so-called admissibility problem. Of course, a necessary condition is provided by the second law of thermodynamics which states that the entropy must not decrease during adiabatic changes of state. For perfect gases, this requirement is also a sufficient condition to rule out “impossible” shocks. For fluids with an arbitrary equation of state and/or situations where in addition to thermoviscous effects, dispersive effects also come into play this is not the case in general. The selection of physically admissible solutions is then found to be a more delicate matter and may result in new types of shocks which differ distinctively from their classical counterparts and, therefore, are termed non-classical shocks.