  We introduce bridge trisections of knotted surfaces in the four-sphere. This
description is inspired by the work of Gay and Kirby on trisections of
four-manifolds and extends the classical concept of bridge splittings of links
in the three-sphere to four dimensions. We prove that every knotted surface in
the four-sphere admits a bridge trisection (a decomposition into three simple
pieces) and that any two bridge trisections for a fixed surface are related by
a sequence of stabilizations and destabilizations. We also introduce a
corresponding diagrammatic representation of knotted surfaces and describe a
set of moves that suffice to pass between two diagrams for the same surface.
Using these decompositions, we define a new complexity measure: the bridge
number of a knotted surface. In addition, we classify bridge trisections with
low complexity, we relate bridge trisections to the fundamental groups of
knotted surface complements, and we prove that there exist knotted surfaces
with arbitrarily large bridge number.
