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We study properties of Engel elements in weakly branch groups, lying in thegroup of automorphisms of a spherically homogeneous rooted tree. Moreprecisely, we prove that the set of bounded left Engel elements is alwaystrivial in weakly branch groups. In the case of branch groups, the existence ofnon-trivial left Engel elements implies that these are all $p$-elements andthat the group is virtually a $p$-group (and so periodic) for some prime $p$.We also show that the set of right Engel elements of a weakly branch group istrivial under a relatively mild condition. Also, we apply these results towell-known families of weakly branch groups, like the multi-GGS groups.