Twisted quadratic differentials, also known as dilation surfaces, are geometric structures that are in a way a generalization of translation surfaces. One way to define a dilation surface is as a collection of polygons with sides identified by translations and dilations by nonzero real factors, whereas translation surfaces only allow side identifications by translations. This small generalization is enough to introduce interesting new dynamical behaviors on dilation surfaces that do not occur for translation surfaces. In this talk, we will introduce dilation surfaces and discuss some of these new and interesting dynamical behaviors that can occur on them, present a construction for some dilation surfaces with large affine automorphism groups, and end by discussing directions for future research about the Teichmuller dynamics of dilation surfaces.