The field was defined by this seminal paper:

Michael Kapovich and John Millson
"On the moduli space of polygons in the Euclidean plane."
*J. Differential Geom.* Volume 42, Number 1 (1995), 133-164.

The topic is now covered to various depths in textbooks, e.g.,
in Discrete and Computational Geometry:

(

*Responding to the OP's emphasis on simple (non-self-crossing) polygons*:)
There has been intense study of the configuration space of simple polygons
with the same edge lengths in the same sequence. Thus the edges can be
viewed as rigid links, joined at vertices by universal joints.
The major result here, by Connolly, Demaine, and Rote, is that this space is
connected: one can continuously "morph" between any two instances of a polygon,
maintaining simplicity throughout. The proof shows that any polygon can
be convexified, and that any two convex instances of the same polygon can be connected.
The "doubled tree" example below was at one time thought a possible counterexample:

_{Animation due to Erik Demaine}
Robert Connelly, Erik D. Demaine, and Günter Rote, “Straightening Polygonal Arcs and Convexifying Polygonal Cycles”, *Discrete & Computational Geometry*, volume 30, number 2, September 2003, pages 205–239. (Author link)

More information on this topic may be found in the book *Geometric Folding Algorithms: Linkages, Origami, Polyhedra*.