  Engel groups and Engel elements became popular in 50s. We consider in the
paper the more general nil-groups and nil-elements in groups. All these notions
are related to nilpotent groups and nilpotent radicals in groups. These notions
generate problems which are parallel to Burnside problems for periodic groups.
  The first three theorems of the paper are devoted to nil-groups and Engel
groups, while the other results are connected with the further generalizations.
These generalizations extend the theory to solvable groups and solvable
radicals in groups. The paper has two parts. The first one (sections 2-4) deals
with old ideas, while the second one (sections 5-9) is devoted to
generalizations.
