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In this talk we make a survey of some open problems and recent results about bounding the Fitting height and $p$-length of nite soluble groups. In many problems of nite group theory, nowadays the classication greatly facilitates reduction to soluble groups. Bounding their Fitting height or $p$-length can be regarded as further reduction to nilpotent groups. This is usually achieved by methods of representation theory, such as Cliord's theorem or theorems of Hall-Higman type. In some problems, it is the case of nilpotent groups where open questions remain, in spite of great successes achieved, in particular, by using Lie ring methods. But there are also important questions that still require reduction to nilpotent groups, which is the focus of this talk. As examples, we discuss nite groups with xed-point-free and almost fixed-point-free automorphisms, generalizations of the Restricted Burnside Problem, and coset identities appearing in the study of pronite groups. Time permitting, we shall mention the open problem of bounding the Fitting height in the study of the analogue of the Restricted Burnside Problem for Moufang loops.