  We study irreducible holomorphic symplectic manifolds deformation equivalent
to Hilbert schemes of points on a $K3$ surface and admitting a non-symplectic
involution. We classify the possible discriminant forms of the invariant and
anti-invariant lattice for the action of the involution on cohomology, and
explicitly describe the lattices in the cases where the invariant has small
rank. We also give a modular description of all $d$-dimensional families of
manifolds of $K3^{[n]}$-type with a non-symplectic involution for $d\geq 19$
and $n\leq 5$, and provide examples arising as moduli spaces of twisted sheaves
on a $K3$ surface.
