We classify the maximal irreducible periodic subgroups of PGL(q; F), where F is a field of positive characteristic p transcendental over its prime subfield, q not equal p is prime, and F-x has an element of order q. That is, we construct a list of irreducible subgroups G of GL(q; F) containing the centre Fx1(q) of GL(q; F), such that G/Fx1(q) is a maximal periodic subgroup of PGL(q; F), and if H is another group of this kind then H is GL(q; F)-conjugate to a group in the list. We give criteria for determining when two listed groups are conjugate, and show that a maximal irreducible periodic subgroup of PGL(q; F) is self-normalising.