A survey of bridge trisections, I

In 2015, Jeff Meier and I introduced bridge trisections of knotted surfaces in the 4-sphere, a natural adaptation of Gay-Kirby trisections.  Bridge trisections also give rise to tri-plane diagrams, a new way of doing knotted surface theory that reflects classical knot theory.  In later work, we extended the definition of bridge trisections to surfaces in arbitrary 4-manifolds.  We will survey some examples, current results, and open questions.