The generalised wreath product of permutation groups, due to Dixon, Fournelle and Silcock, is studied in this thesis. Under nice conditions this turns out to be a good generalisation of the permutational wreath product. We will explain precisely what we mean by nice. The centre of the generalised wreath product is determined and we look at the centraliser of certain elements of the group. The remainder of the thesis is concerned with looking to answer the question: given a class X, can we find necessary and sufficient conditions for the generalised wreath product to lie in X? We consider the class of abelian groups; nilpotent groups; locally nilpotent groups; ZA groups; residually nilpotent groups; locally boundedly nilpotent groups; bounded Engel groups; soluble groups; and locally soluble groups.