  The aim of this paper is to investigate in which sense, for $n\geq 3$,
$n$-Lie algebras admit universal enveloping algebras. There have been some
attempts at a construction (see [10] and [5]) but after analysing those we come
to the conclusion that they cannot be valid in general. We give counterexamples
and sufficient conditions.
  We then study the problem in its full generality, showing that universality
is incompatible with the wish that the category of modules over a given $n$-Lie
algebra $L$ is equivalent to the category of modules over the associated
algebra $U(L)$. Indeed, an associated algebra functor $U \colon
\text{$n$-}\mathsf{Lie}_{\mathbb{K}} \to \mathsf{Alg}_{\mathbb{K}}$ inducing
such an equivalence does exist, but this kind of functor never admits a right
adjoint.
  We end the paper by introducing a (co)homology theory based on the associated
algebra functor $U$.
