  We report on extensive numerical simulations on the Bak-Sneppen model in high
dimensions. We uncover a very rich behavior as a function of dimensionality.
For d>2 the avalanche cluster becomes fractal and for d \ge 4 the process
becomes transient. Finally the exponents reach their mean field values for
d=d_c=8, which is then the upper critical dimension of the Bak Sneppen model.
