  Algorithmic computation in polynomial rings is a classical topic in
mathematics. However, little attention has been given to the case of rings with
an infinite number of variables until recently when theoretical efforts have
made possible the development of effective routines. Ability to compute relies
on finite generation up to symmetry for ideals invariant under a large group or
monoid action, such as the permutations of the natural numbers. We summarize
the current state of theory and applications for equivariant Gr\"obner bases,
develop several algorithms to compute them, showcase our software
implementation, and close with several open problems and computational
challenges.
