  We study the rigidity and flexibility of symplectic embeddings of simple
shapes. It is first proved that under the condition $r_n^2 \le 2 r_1^2$ the
symplectic ellipsoid $E(r_1, ..., r_n)$ with radii $r_1 \le ... \le r_n$ does
not embed in a ball of radius strictly smaller than $r_n$. We then use
symplectic folding to see that this condition is sharp and to construct some
nearly optimal embeddings of ellipsoids and polydiscs into balls and cubes. It
is finally shown that any connected symplectic manifold of finite volume may be
asymptotically filled with skinny ellipsoids or polydiscs.
