The moduli space of holomorphic one-forms on a closed surface of genus $g$ has a natural piecewise linear structure and carries an action of GL(2,R). One can investigate the geometry and dynamics of an individual flat surface by studying its orbit under this linear group action. I will discuss several open problems regarding the properties of these orbit closures. This talk is based on joint work with Alex Eskin, Amir Mohammadi and Alex Wright.