A symplectic embedding of a disjoint union of domains into a symplectic manifold M is said to be of Kahler type (respectively tame) if it is holomorphic with respect to some (not a priori fixed) integrable complex structure on M which is compatible with (respectively tamed by) the symplectic form. I'll discuss when Kahler-type embeddings of disjoint unions of balls into a closed symplectic manifold exist and when two such embeddings can be mapped into each other by a symplectomorphism. If time permits, I'll also discuss the existence of tame embeddings of balls, polydisks and parallelepipeds into tori and K3 surfaces.   This is a joint work with M.Verbitsky.