  This work is concerned with the quasineutral limit of the one-dimensional
Vlasov-Poisson equation, for initial data close to stationary homogeneous
profiles. Our objective is threefold: first, we provide a proof of the fact
that the formal limit does not hold for homogeneous profiles that satisfy the
Penrose instability criterion. Second, we prove on the other hand that the
limit is true for homogeneous profiles that satisfy some monotonicity
condition, together with a symmetry condition. We handle the case of
well-prepared as well as ill- prepared data. Last, we study a stationary
boundary-value problem for the formal limit, the so-called quasineutral Vlasov
equation. We show the existence of numerous stationary states, with a lot of
freedom in the construction (compared to that of BGK waves for Vlasov-Poisson):
this illustrates the degeneracy of the limit equation.
