In this thesis we find deep results on the structure of normal right n-Engel subgroups that are contained in some term of the upper central series of a group. We start with some known results, and one new result, on the structure of locally nilpotent n-Engel groups. These are closely related to the solution of the restricted Burnside problem. We also give specific details of the structure of 2-Engel and 3-Engel groups in the context of these results. The main idea of this thesis is to generalise these results to apply to normal upper central right n-Engel subgroups. We also consider the special case of locally finite p-groups and again generalise some deep results on the structure of n-Engel such groups to apply to right n-Engel subgroups. For each of the theorems on right n-Engel subgroups, complete details are given for the case n = 2. Right 3-Engel subgroups have a more complicated structure. For these we prove a Fitting result, for which we exclude the prime 3, and using this we also find a sharp bound on the upper central degree in the torsion-free case. In fact we only need to exclude the primes 2, 3 and 5 for this result. This gives some further information on the structure of right 3-Engel subgroups in the context of the main theorems.