For a set of primes \(\pi\), the concept of Engel conditions for finite groups is extended. For example an element \(g\) is a weakly right \(\pi\)-Engel element if for each \(\pi'\)-element \(x\) in the group \(G\) there is a positive integer \(n\) such that the \((n+1)\)-commutator \([x,g,\dots,g]\) is a \(\pi\)-element. This is extended further to a right (strongly right) \(\pi\)-Engel element together with the corresponding conditions for left rather than right. These conditions are closely identified with the existence of groups with either normal \(\pi\)-complements or to \(\pi\)-nilpotent groups. Representative results are these. If \(2\notin\pi\), then a group \(G\) has a normal \(\pi\)-complement if and only if each \(\pi'\)-element \(x'\in G\) is a weakly right \(\pi'\)-Engel element. On the other hand a group \(G\) has a normal Sylow 2-subgroup if and only if each 2-element of \(G\) is a weakly right 2-Engel element. The authors prove that a group \(G\) is \(\pi\)-nilpotent, that is if \(G\) is \(p\)-nilpotent for each \(p\in\pi\), if and only if \(G\) has a nilpotent Hall \(\pi\)-subgroup and the quotient \({\mathcal N}_G(L)/{\mathcal C}_G(L)\) is a \(\pi\)-group for each \(\pi\)-subgroup \(L\) in \(G\). Furthermore the following statements are equivalent: (i) \(G\) is a \(\pi\)-nilpotent group. (ii) Each \(\pi\)-element of \(G\) is a left \(\pi'\)-Engel element. (iii) For any \(x\in G\) of order \(p\in\pi\), or of order \(4\) if \(2\in\pi\), \(x\) is a weakly left \(p'\)-Engel element. In addition if \(G\) is a quaternion-free group and for each element \(x\in G\) such that \(|x|\in\pi\), \(x\) is a left \(\pi'\)-Engel element, then \(G\) is a \(\pi\)-nilpotent group. The theorems and propositions from which these corollaries evolve are of independent interest.