Curves with Caputo's fractional derivative

İn this tesis; a new definition of the fractional curvature of plane curves is given bu means of the Caputo derivative of the fractional derivative. This new curvature is based on the behavior of the orthogonal projection of the integer derivative of the fractional curvature vector of the curve on the vector normal to the curve. The aim of the thesis is to show that our new fractional curvature belongs to the intrinsic geometry of the curve. Another aim of the thesis is to investigate planar congruent curves according to the Caputo fractional derivative. Here, the tangent and normal vector according to the Caputo fractional derivative are also redefined. Thus, congruent curves are analyzed according to these newly defined vectors.