I define a $K3$ surface as a smooth complex manifold of dimension two which is simply-connected and such that the canonical bundle is trivial.
I know that two $K3$ surfaces are always deformation equivalents and I know that $K3$ surfaces are Kähler. Conversely, if a deformation $X$ of a $K3$ surface is Kähler, then Hodge structure is preserved so $X$ is again simply-connected, and the symplectic form on the $K3$ surface extends to $X$ so the canonical bundle of $X$ is trivial too. Hence $X$ is a $K3$ surface.
How about non-Kähler deformations of a $K3$ surface? If they exists they aren't $K3$ surfaces, but do they exists?