In this thesis we investigate the structure of locally nilpotent 5-Engel p-groups. We show that for p > 7, locally nilpotent 5-Engel p-groups have class at most 10. This is a global theorem, where the result is not dependent on the number of generators of the group. The proof uses new and established Lie methods and a custom C++ implementation of an algorithm that constructs minimal generating sets and structure constants of multi- graded Lie algebras in a variety defined by three multilinear relations, which hold in the Lie rings associated with 5-Engel p-groups. We obtain our results by calculating in the set Q(p) = {~ I x E Z, yE Z+, Y # 0 modulo any p f/. p} (where p is a set of excluded primes and x, y are arbitrarily large integers), as well as the fields Zp, p prime. We introduce several reduction theorems, making the result possible. We also present results about the normal closure of elements in these groups. We use a Higman reduction theorem and the same custom C++ program to show that locally nilpotent 5-Engel p-groups, p 2: 5, are Fitting, with Fitting degree at most 4 if p > 7, at most 5 if p = 7 and at most 6 if p = 5. These results are best possible.