In 1994, V.I. Arnold introduced three invariants St, J+, J- for generic immersions of a circle into the plane. He formulated four conjectures about the upper and
lower bounds of these invariants. All of these conjectures had been proved by 1998. The main part of this thesis is a survey on results emerged after the work of V.I.
Arnold. In the last chapter, I introduce a polynomial invariant for generic based immersions in the plane and calculate the polynomials of based tree-like curves. This
construction seems to be new.

Keywords : plane curves, generic immersions, the Arnold invariants, tree-like curves, Gauss diagrams, a polynomial invariant.