Fair division is the problem of dividing a set of items among $n$ agents in a fair manner. When the items are indivisible, the classical fairness solution concepts of {\em envy-freeness} and {\em proportionality} are rendered inapplicable. In this talk, I will survey recent advances on some of the strongest relaxations of these two concepts, namely EFX and MMS/APS, and their connections to other areas of theory CS, such as extremal combinatorics, probabilistic methods, and the Santa Clause problem.