We show how to define domains of processes, which arise in the denotational 
semantics of concurrent languages, using hypersets, i.e. non-wellfounded sets.
In particular we discuss how to solve recursive equations involving
set-theoretic operators within {\em hyperuniverses} with atoms.
Hyperuniverses are transitive sets which carry a uniform topological structure 
and include as a clopen subset their exponential space (i.e. the set of their
closed subsets) with the exponential uniformity.
This approach allows to solve many recursive domain equations of processes
which cannot be even expressed in standard Zermelo-Fraenkel Set Theory, e.g.
when the functors involved have negative occurrences of the argument. Such
equations arise in the semantics of concurrrent programs in connection with
function spaces and higher order assignment. Finally, we briefly compare our
results to those which make use of complete metric spaces, due to de Bakker,
America and Rutten.