I made a top-level parent comment about a result from March 2017. Otherwise, things that either will become part of history, or suggest directions for work that will:

* Green-Tao theorem, and lots of other work involving Terry Tao

* Yitang Zhang's work on prime gaps, and subsequent improvements

* Bhargava and collaborators' work on elliptic curves

* The introduction of algebraic topology methods into algebraic geometry (aka "simplicial"/"derived" stuff, see Jacob Lurie)

* Homotopy type theory

* Mochizuki/ABC stuff

* Peter Scholze's work in arithmetic geometry