  We show that there is a stable homotopy theory of profinite spaces and use it
for two main applications. On the one hand we construct an \'etale topological
realization of the stable motivic homotopy theory of smooth schemes over a base
field of arbitrary characteristic in analogy to the complex realization functor
for fields of characteristic zero. On the other hand we get a natural setting
for \'etale cohomology theories. In particular, we define and discuss an
\'etale topological cobordism theory for schemes. It is equipped with an
Atiyah-Hirzebruch spectral sequence starting from \'etale cohomology. Finally,
we construct maps from algebraic to \'etale cobordism and discuss algebraic
cobordism with finite coefficients over an algebraically closed field after
inverting a Bott element.
