  McDuff and Schlenk have recently determined exactly when a four-dimensional
symplectic ellipsoid symplectically embeds into a symplectic ball. Similarly,
Frenkel and M\"uller have recently determined exactly when a symplectic
ellipsoid symplectically embeds into a symplectic cube. Symplectic embeddings
of more complicated structures, however, remain mostly unexplored. We study
when a symplectic ellipsoid $E(a,b)$ symplectically embeds into a polydisc
$P(c,d)$. We prove that there exists a constant $C$ depending only on $d/c$
(here, $d$ is assumed greater than $c$) such that if $b/a$ is greater than $C$,
then the only obstruction to symplectically embedding $E(a,b)$ into $P(c,d)$ is
the volume obstruction. We also conjecture exactly when an ellipsoid embeds
into a scaling of $P(1,b)$ for $b$ greater than or equal to $6$, and conjecture
about the set of $(a,b)$ such that the only obstruction to embedding $E(1,a)$
into a scaling of $P(1,b)$ is the classical volume. Finally, we verify our
conjecture for $b = \frac{13}{2}$.
