  Free probability analogs of the basics of extreme-value theory are obtained,
based on Ando's spectral order. This includes classification of freely
max-stable laws and their domains of attraction, using ``free extremal
convolutions'' on the distributions. These laws coincide with the limit laws in
the classical peaks-over-threshold approach. A free extremal projection-valued
process over a measure-space is constructed, which is related to the free
Poisson point process.
