I will discuss some symplectic embedding problems and their relation to questions in complex geometry.  There is an interesting class of embedding problems in which the domain is a four-dimensional ellipsoid and the target is the projective plane or other rational  surface.  The embedding obstructions turn out to be related to the 
rational unicuspidal curves in these surfaces. I will also explain how to see these curves via the almost toric structures supported by the target. This is joint work with Kyler Siegel.