A classic result, due to McDuff and Schlenk, asserts that the function that encodes when a four-dimensional symplectic ellipsoid can be embedded into a four-dimensional ball has a remarkable structure: the function has infinitely many corners, determined by the odd-index Fibonacci numbers, that fit together to form an infinite staircase. The work of McDuff and Schlenk has recently led to considerable interest in understanding when the ellipsoid embedding function for other symplectic 4-manifolds is partly described by an infinite staircase.  In this talk we will discuss a general framework for analyzing this question for a large family of targets, and in particular give an obstruction to the existence of an infinite staircase that experimentally seems strong. We will then look at the special case of rational convex toric domains / closed symplectic toric manifolds, for which we prove the existence of six families of targets with infinite staircases that are distinguished by the fact that their moment polygon is reflexive. The proof uses, among other tools, almost toric fibrations -- see also the second of the ellipsoid day talks. Finally, we conjecture that these six families constitute a complete answer to the question of existence of infinite staircase. This conjecture has been verified in the case when the target is an ellipsoid -- see the third of the ellipsoid day talks. This is based on joint work of Dan Cristofaro-Gardiner, Tara Holm, Alessia Mandini, and Ana Rita Pires.