For a given positive integer
n
n
and a given prime number
p
p
, let
r
=
r
(
n
,
p
)
r=r(n,p)
be the integer satisfying
p
r
−
1
>
n
≤
p
r
p^{r-1}>n\leq p^{r}
. We show that every locally finite
p
p
-group, satisfying the
n
n
-Engel identity, is (nilpotent of
n
n
-bounded class)-by-(finite exponent) where the best upper bound for the exponent is either
p
r
p^{r}
or
p
r
−
1
p^{r-1}
if
p
p
is odd. When
p
=
2
p=2
the best upper bound is
p
r
−
1
,
p
r
p^{r-1},p^{r}
or
p
r
+
1
p^{r+1}
. In the second part of the paper we focus our attention on
4
4
-Engel groups. With the aid of the results of the first part we show that every
4
4
-Engel
3
3
-group is soluble and the derived length is bounded by some constant.