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The question of whether one symplectic manifold can be embedded into another can be very subtle, even for simple domains in C^n. For example, McDuff and Schlenk computed when a four-dimensional ellipsoid can be symplectically embedded into a ball, and found that if the ellipsoid is close to round, the answer is given by an ``infinite staircase" determined by the odd-index Fibonacci numbers and the Golden Mean. Later, McDuff was able to find a purely combinatorial criterion for determining when one four-dimensional ellipsoid can be embedded into another; in higher dimensions, however, little is known. I will report on joint work with McDuff and Hind studying higher dimensional ellipsoid embeddings. We show that a version of the McDuff-Schlenk result holds in all dimensions, and we also find "ghost" obstructions that are hidden in dimension four, but active otherwise.