Classical curve theory in normed planes

The classical theory of individual classes of planar curves is a well-known field between Elementary, Differential, and Algebraic Geometry. With the present expository paper we want to point out the fact that an extension of this field to two-dimensional real Banach spaces, also called normed or Minkowski planes, is still missing. We want to show that until now only a few topics from this natural and rich geometric field were extended to Minkowski planes, and that, moreover, even in these directions only partial results exist. We present these known results, give open problems and show possible directions of future research. It is our goal to verify that classical curve theory in Minkowski planes can be nicely developed to become a very wide and interesting research subject in the spirit of different modern fields, like Differential Geometry, Functional Analysis, Computational Geometry and related directions.