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Here we investigate meaningful families of vector bundles on a very generalpolarized $K3$ surface $(X,H)$ and on the corresponding Hyper--Kaehler varietygiven by the Hilbert scheme of points $X^{[k]}:= {\rm Hilb}^k(X)$, for anyinteger $k \geqslant 2$. In particular, we prove results concerning bigness andstability of such bundles.  First, we give conditions on integers $n$ such that the twist of the tangentbundle of $X$ by the line bundle $nH$ is big and stable on~$X$; we then prove asimilar result for a natural twist of the tangent bundle of $X^{[k]}$. Next, weprove global generation, bigness and stability results for tautological bundleson $X^{[k]}$ arising either from line bundles or from Mukai-Lazarsfeld bundles,as well as from Ulrich bundles on $X$, using a careful analysis on Segreclasses and numerical computations for $k = 2, 3$.

Some families of big and stable bundles on $K3$ surfaces and on their Hilbert schemes of points