  Epigroups are semigroups equipped with an additional unary operation called
pseudoinversion. Each finite semigroup can be considered as epigroup. We prove
the following theorem announced by Zhil'tsov in 2000: the equational theory of
the class of all epigroups coincides with the equational theory of the class of
all finite epigroups and is decidable. We show that the theory is not finitely
based but provide a transparent infinite basis for it.
