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Let n be a positive integer. We say that a group G is an (n + ½)-Engel group if it satisfies the law [x, ny, x ] = 1. The variety of (n + ½)-Engel groups lies between the varieties of n-Engel groups and (n + 1)-Engel groups. In this paper, we study these groups, and in particular, we prove that all (4 + ½)-Engel {2,3}-groups are locally nilpotent. We also show that if G is a (4 + ½)-Engel p-group, where p ≥ 5 is a prime, then Gp is locally nilpotent. [ABSTRACT FROM AUTHOR]