  In this paper we explore the nature of self-similar solutions of the Curve
Shortening Flow and the Vortex Filament Equation, also known as the Binormal
Flow. We explore some of their fundamental conservation properties and describe
the behavior of their self-similar solutions. For Curve Shortening Flow we
mainly expose the results of Huisken, Grayson, and Halldorsson concerning the
equation's basic properties and self-similar solutions in the plane. For the
Vortex Filament Equation we present the results by Banica and Vega, Arms and
Hama, and Hasimoto. We also derive the evolution equations of the normal,
binormal and tangent vectors in the Frenet frame for the vortex filament as
well as those of curvature and torsion. We give a proof that circles are the
only planar translating self-similar solutions and also derive a system of
ordinary differential equations that govern the behavior for rotating
